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WINDOW  SHAMS. 


Irving  Stringham 


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ELEMENTS 


PLANE    AND    SPHERICAL 

TRIGONOMETEY, 

WITH   THEIR   APPLICATIONS  TO 

MENSURATION,  SURVEYING,  AND 
NAVIGATION. 

BY    ELIAS    LOOMIS,    LL.D., 

PROFESSOR    OF  NATURAL  PHILOSOPHY  AND  ASTRONOMY  IN  YALE   COLLEGE,  AND   AUTHOR   OP 

A  "COURSE  OF  MATHEMATICS." 
TWENTY-FIFTH    EDITION. 


NEW    YORK: 

HARPER    &    BROTHERS,    PUBLISHERS, 

FRANKLIN    SQUARE. 

1868. 


*J**+*<^ 


. 

M,  LOOMIS'S  MATHEMATICAL  SERIES, 

PUBLISHED  BY  , 

HARPER  &  BROTHERS,  NEW  YOEK. 


ELEMENTARY  ARITHMETIC.  Elements  of  Arithmetic: 
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PREFACE. 


THE  following  treatise  constitutes  the  third  volume  of  m 
course  of  Mathematics  designed  for  colleges  and  high  schools, 
and  is  prepared  upon  substantially  the  same  model  as  the  works 
on  Algebra  and  Geometry.  It  does  not  profess  to  embody 
every  thing  which  is  known  on  the  subject  of  Trigonometry, 
but  it  contains  those  principles  which  are  most  important  on 
account  of  their  applications,  or  their  connection  with  other 
parts  of  a  course  of  mathematical  study.  The  aim  has  been 
to  render  every  principle  intelligible,  not  by  the  repetition  of 
superfluous  words,  but  by  the  use  of  precise  and  appropriate 
language.  Whenever  it  could  conveniently  be  done,  the  most 
important  principles  have  been  reduced  to  the  form  of  theorems 
or  rules,  which  are  distinguished  by  the  use  of  italic  letters, 
and  are  designed  to  be  committed  to  memory.  The  most  im- 
portant instruments  used  in  Surveying  are  fully  described,  and 
are  illustrated  by  drawings. 

The  computations  are  all  made  by  the  aid  of  natural  num- 
bers, or  with  logarithms  to  six  places ;  and  by  means  of  the 
accompanying  tables,  such  computations  can  be  performed 
with  gre^it  facility  and  precision  This  volume,  having  been 
used  by  several  successive  classes,  has  been  subjected  to  the 
severest  scrutiny,  and  the  present  edition  embodies  all  the  al- 
terations which  have  been  suggested  by  experience  in  the  re- 
citation rcorn. 


812239 


4 


CONTENTS 


BOOK  I. 

THE  NATURE  AND  PROPERTIES  OF  LOGARITHMS. 

ft  f 

Nature  of  Logarithms 7 

Description  of  the  Table  of  Logarithms 9 

Multiplication  by  Logarithms 1 

Division  by  Logarithms H 

Involution  by  Logarithms 17 

Evolution  by  Logarithms 17 

Proportion  by  Logarithms 18 

BOOK  II. 

PLANE  TRIGONOMETRY. 

Sines,  Tangents,  Secants,  &c.,  defined 20 

Explanation  of  the  Trigonometrical  Tables 23 

To  find  Sines  and  Tangents  of  small  Arcs 29 

Solutions  of  Right-angled  Triangles 32 

Solutions  of  Oblique-angled  Triangles 36 

Instruments  used  in  Drawing 42 

Geometrical  Construction  of  Triangles 40 

Values  of  the  Sines,  Cosines,  &c.,  of  certain  Angles « 48 

Trigonometrical  Formula) 52 

Computation  of  a  Table  of  Sines,  Cosines,  &c 57 

BOOK  III. 

MENSURATION  OF  SURFACES  AND  SOLIDS. 

Areas  of  Figures  bounded  by  Right  Lines 59 

Area  of  a  Regular  Polygon G4 

Quadrature  of  the  Circle  and  its  Parts 6G 

Mensuration  of  Solids 71 

Rail-way  Excavations  or  Embankments 77 

Regular  Polyedrons 81 

The  three  Round  Bodies 84 

Area  of  a  Spherical  Triangle 8S 

BOOK  IV. 

SURVEYING. 

Definitions OC 

Instruments  for  measuring  Angles 91 


vi  CONTENT  s. 

Pag* 

Explanation  of  the  Vernier 94 

Description  of  the  Theodolite 9a 

Heights  and  Distances 97 

The  Determination  of  Areas 3  03 

Plotting  a  Survey ^104 

The  Traverse  Table 106 

Trffindthe  Area  of  a  Field 109 

Trigonometrical  Surveys 114 

Variation  of  the  Needle 117 

Leveling 119 

Topographical  Maps 123 

Setting  out  Rail-way  Curves 127 

Surveying  Harbors 130 

The  Plane  Table 132 

To  determine  the  Depth  of  Water 133 

BOOK  V. 

NAVIGATION. 

Definitions,  &c 135 

Plane  Sailing 138 

Traverse  Sailing 141 

Parallel  Sailing 144 

Middle  Latitude  Sailing 146 

Mercator's  Sailing 149 

Nautical  Charts 153 

BOOK  VI. 

SPHERICAL  TRIGONOMETRY. 

Right-angled  Spherical  Triangles 155 

Napier's  Rule  of  the  Circular  Parts 158 

Examples  of  Right-angled  Triangles 160 

Oblique-angled  Spherical  Triangles 163 

Examples  of  Oblique-angled  Triangles 165 

Trigonometrical  Formulas 171 

Sailing  on  an  Arc  of  a  Great  Circle I7fi 


TEIGOIOMET11Y. 


*  BOOK  I. 

THE  NATURE  AND  PROPERTIES  OF  LOGARITHMS. 

ARTICLE  1.  Logarithms  are  numbers  designed  to  diminish 
the  labor  of  Multiplication  and  Division,  by  substituting  in  their 
stead  Addition  and  Subtraction.  All  numbers  are  regarded  as 
powers  of  some  one  number,  which  is  called  the  base  of  the 
system  ;  and  the  exponent  of  that  power  of  the  base  which  is 
equal  to  a  given  number,  is  called  the  logarithm  of  that  number. 

The  base  of  the  common  system  of  logarithms  (called,  from 
their  inventor,  Briggs'  logarithms)  is  the  number  10.  Hence 
all  numbers  are  to  be  regarded  as  powers  of  10.  Thus,  since 
10°= 1,  0  is  the  logarithm  of  1  in  Briggs'  system; 

10'=10,         1      "  "  10  «  " 

10S=100,       2      "  "  100  "  " 

103=1000,     3      "  «  1000  «  « 

104=10000,  4      "  "  10,000  «  « 

&c.,  &o.,  &c. ; 

whence  it  appears  that,  in  Briggs'  system,  the  logarithm  ol 
every  number  between  1  and  10  is  some  number  between  0 
and  1,  i.  e.,  is  a  proper  fraction.  The  logarithm  of  every  num- 
ber between  10  and  100  is  some  number  between  1  and  2,  i.  e., 
is  1  plus  a  fraction.  The  logarithm  of  every  number  between 
100  and  1000  is  some  number  between  2  and  3,  i.  e.,  is  2  plus 
a  fraction,  and  so  on. 

(2.)  The  preceding  principles  may  bo  extended  to  fractions 
by  means  of  negative  exponents.     Thus,  since 
10— J  =  0.1,         - 1  is  the  logarithm  of  0.1       in  Briggs'  system ; 
10-2=0.01,      -2     "  «  0.01  «  « 

10-3  =0.001,    -3     "  "  0.001  "  « 

10-4=0.0001  -4     «  v<"  0.0001       .  "  « 

&c.,  &c.,  &o. 


TRIGONOMETRY. 


-»      • 

Hence  it'  appears  'that  the  logarithm  of  every  number  between 
1  and  0  1  is  some  number  between  0  and  —1,  or  may  be  rep 
resented  by  —  1  plus  a  fraction  ;  the  logarithm  of  every  num- 
ber between  0.1  and  .01  is  some  number  between  —1  and  —2. 
or  may  be  represented  by  —2  plus  a  fraction;  the  logarithm 
of  every  number  between  .01  and  .001  is  some  number  be- 
tween —2  and  —3,  or  is  equal  to  —3  plus  a  fraction,  and 
so  on. 

The  logarithms  of  most  numbers,  therefore,  consist  of  an  in 
teger  and  a  fraction.  The  integral  part  is  called  the  charac 
t  eristic,  and  may  be  known  from  the  following 

RULE. 

The  characteristic  of  the  logarithm  of  any  number  greater 
than  unity,  is  one  less  than  the  number  of  integral  figures  in 
the  given  number. 

Thus  the  logarithm  of  297  is  2  plus  a  fraction  ;  that  is,  the 
characteristic  of  the  logarithm  of  297  is  2,  which  is  one  less 
than  the  number  of  integral  figures.  The  characteristic  of  the. 
logarithm  of  5673.29  is  3  ;  that  of  73254.1  is  4,  &o. 

The  characteristic  of  the  logarithm  of  a  decimal  fraction 
is  a  negative  number,  and  is  equal  to  the  number  of  places  by 
which  its  first  significant  figure  is  removed  from  the  place 
of  units. 

Thus  the  logarithm  of  .0046  is  —3  plus  a  fraction  ;  that  is, 
the  characteristic  of  the  logarithm  is  —3,  the  first  significant 
figure,  4,  being  removed  three  places  from  units. 

(3.)  Since  powers  of  the  same  quantity  are  multiplied  by 
Adding  their  exponents  (Alg.,  Art.  50), 

The  logarithm  of  the  product  of  two  or  more  factors  is 
pqual  to  the  sum  of  the  logarithms  of  those  factors. 

Hence  we  see  that  if  it  is  required  to  multiply  two  or  more 
numbers  by  each  other,  we  have  only  to  add  their  logarithms  : 
the  sum  will  be  the  logarithm  of  their  product.  We  then  look 
in  the  table  for  the  number  answering  to  that  logarithm,  in 
order  to  obtain  the  required  product. 

Also,  since  powers  of  the  same  quantity  are  divided  by  sub- 
tracting their  exponents  (Alg.,  Art.  66), 

The  logarithm  of  the  quotient  of  one  number  divided  ly  an- 


LOGARITHMS.  9 

>ther,  is  equal  to  the  difference  of  the  logarithms  of  those, 
numbers. 

Hence  we  see  that  if  we  wish  to  divide  one  number  by  an- 
other,  we  have  only  to  subtract  the  logarithm  of  the  divisoi 
from  that  of  the  dividend  ;  the  difference  will  be  the  logarithm 
of  their  quotient. 

(4.)  Since,  in  Briggs'  system,  the  logarithm  of  10  is  1,  if 
any  number  be  multiplied  or  divided  by  10,  its  logarithm,  will 
be  increased  or  diminished  by  1 ;  and  as  this  is  an  integer,  it 
will  only  change  the  characteristic  of  the  logarithm,  without 
affecting  the  decimal  part.  Hence 

The  decimal  part  of  the  logarithm  of  any  number  is  the 
same  as  that  of  the  number  multiplied  or  divided  by  10,  100, 
1000,  &o. 

Thus,  the  logarithm  of  65430  is  4.815777  ; 

"  "  6543  is  3.815777; 

"  "  654.3  is  2.815777; 

"  "  65.43          is  1.815777 ; 

"  "  6.543        is  0.815V77; 

"  "  6543      is  1.816777 ; 

"  "  06543    is  2.815777; 

"  "  ,006543  is  3.815777. 

The  minus  sign  is  here  placed  over  the  characteristic,  to 
show  that  that  alone  is  negative,  while  the  decirial  part  of  the 
logarithm  is  positive. 

TABLE  OF  LOGARITHMS. 

(5.)  A  table  of  logarithms  usually  contains  the  logarithms 
of  the  entire  series  of  natural  numbers  from  1  up  to  10,000, 
and  the  larger  tables  extend  to  100,000  or  more.  In  the  smaller 
tables  the  logarithms  are  usually  given  to  five  or  six  decimai 
places  ;  the  larger  tables  extend  to  seven,  and  sometimes  eight 
or  more  places. 

In  the  accompanying  table,  the  logarithms  of  the  first  100 
numbers  are  given  with  their  characteristics  ;  but,  for  all  other 
numbers,  the  decimal  part  only  of  the  logarithm  is  given,  while 
the  characteristic  is  left  to  be  supplied,  according  to  the  rule 
in  Art.  2. 


10  TRIGONOMETRY. 

(6.)  To  find  the  Logarithm  of  any  Number  between  1  and  100 
Look  on  the  first  page  of  the  accompanying  table,  along  the 
column  of  numbers  under  N.,  for  the  given  number,  and  against 
it,  in  the  next  column,  will  be  found  the  logarithm  with  its 
characteristic.  Thus, 

opposite  13  is  1.113943,  which  is  the  logarithm  of  13 ; 
«       65  is  1.812913,  «  "  65. 

To  find  the  Logarithm  of  any  Number  consisting  of  three 

Figures. 

Look  on  one  of  the  pages  of  the  table  from  2  to  20,  along 
the  left-hand  column,  marked  N.,  for  the  given  number,  and 
against  it,  in  the  column  headed  0,  will  be  found  the  decimal 
part  of  its  logarithm.  To  this  the  characteristic  must  be  pre- 
fixed, according  to  the  rule  in  Art.  2.  Thus 
the  logarithm  of  347  will  be  found,  from  page  8,  2.540329 ; 

"  u          871  "  "       18,  2.940018. 

As  the  first  two  figures  of  the  decimal  are  the  same  for  sev 
eral  successive  numbers  in  the  table,  they  are  not  repeated  for 
each  logarithm  separately,  but  are  left  to  be  supplied.  Thus 
the  decimal  part  of  the  logarithm  of  339  is  .530200.  The  first 
two  figures  of  the  decimal  remain  the  same  up  to  347 ;  they 
are  therefore  omitted  in  the  table,  and  are  to  be  supplied. 

To  find  the  Logarithm  of  any  Number  consisting  of  foii'i 

Figures. 

Find  the  three  left-hand  figures  in  the  column  marked  N., 
as  before,  and  the  fourth  figure  at  the  head  of  one  of  the  other 
columns.  Opposite  to  the  first  three  figures,  and  in  the  col- 
umn under  the  fourth  figure,  will  be  found  four  figures  of  the 
logarithm,  to  which  two  figures  from  the  column  headed  0  are 
to  be  prefixed,  as  in  the  former  case.  The  characteristic  must 
be  supplied  according  to  Art.  2.  Thus 

the  logarithm  of  3456  is  3.538574 ; 
"  "          8765  is  3.942752. 

In  several  of  the  columns  headed  1,  2,  3,  &c.,  small  dots  are 
found  in  the  place  of  figures.  This  is  to  show  that  the  two 
figures  which  are  to  be  prefixed  from  the  first  column  have 
changed }  and  th3y  are  to  be  taken  from  the  horizrntal  line  di- 


W... 


LOGARITHMS.  Jl 

rectly  below.     The  place  of  the  dots  is  to  be  supplied  with  ci- 
phers.    Thus 

the  logarithm  of  2045  is  3.310693  ; 
"  "          9777  is  3.990206. 

The  two  leading  figures  from  the  column  0  must  also  be 
taken  from  the  horizontal  line  below,  if  any  dots  have  been 
passed  over  on  the  same  horizontal  line.     Thus 
the  logarithm  of  1628  is  3.211654. 

To  find  the  logarithm  of  any  Number  containing  more  than 

four  Figures. 

(7.)  By  inspecting  the  table,  we  shall  find  that,  within  cer- 
tain limits,  the  differences  of  the  logarithms  are  nearly  propor- 
tional to  the  differences  of  their  corresponding  numbers.    Thus 
the  logarithm  of  7250  is  3.860338  ; 
"  "          7251  is  3.860398 ; 

"  "          -7252  is  3.860458 ; 

7253  is  3.860518. 

Here  the  difference  between  the  successive  logarithms,  called 
the  tabular  difference,  is  constantly  60,  corresponding  to  a  dif- 
ference of  unity  in  the  natural  numbers.  If,  then,  we  sup- 
pose the  logarithms  to  be  proportional  to  their  corresponding 
numbers  (as  they  are  nearly),  a  difference  of  0.1  in  the  num- 
bers should  correspond  to  a  difference  of  6  in  the  logarithms ; 
a  difference  of  0.2  in  the  numbers  should  correspond  to  a  dif- 
ference of  12  in  the  logarithms,  &c.  Hence 

the  logarithm  of  7250.1  must  be  3.860344 ; 
"  "          7250.2         «      3.860350; 

"  «          7250.3         "      3.860356. 

In  order  to  facilitate  the  computation,  the  tabular  difference 
is  inserted  on  page  16  in  the  column  headed  D.,  and  the  pro- 
portional part  for  the  fifth  figure  of  the  natural  number  is  given 
at  the  bottom  of  the  page.  Thus,  when  the  tabular  difference 
is  60,  the  corrections  for  .1,  .2,  .3,  ,&c.,  are  seen  to  be  6,  12, 
18,  &c. 

If  the  given  number  was  72501,  the  characteristic  of  its  log- 
arithm would  be  4,  but  the  decimal  part  would  be  the  same  as 
for  7250.1. 

If  it  were  required  to  find  the  correction  for  a  sixth  figure 


12  TRIGONOMETRY. 

in  the  natural  number,  it  is  readily  obtained  from  the  Propor- 
tional Parts  in  the  table.  The  correction  for  a  figure  in  the 
sixth  place  must  be  one  tenth  of  the  correction  for  the  same 
figure  if  it  stood  in  the  fifth  place.  Thus,  if  the  correction  for 
.5  is  30,  the  correction  for  .05  is  obviously  3. 

As  the  differences  change  rapidly  in  the  first  part  of  the  ta- 
ble, it  was  found  inconvenient  to  give  the  proportional  parts 
for  each  tabular  difference ;  accordingly,  for  the  first,  seven 
pages,  they  are  only  given  for  the  even  differences,  but  the  pro- 
portional parts  for  the  odd  differences  will  be  readily  found  by 
inspection. 

Required  the  logarithm  of  452789. 

The  logarithm  of  452700  is  5.655810. 

The  tabular  difference  is  96. 

Accordingly,  the  correction  for  the  fifth  figure,  8,  is  77,  and 
for  the  sixth  figure,  9,  is  8.6,  or  9  nearly.  Adding  these  cor- 
rections to  the  number  before  found,  we  obtain  5.655896. 

The  preceding  logarithms  do  not  pretend  to  be  perfectly 
exact,  but  only  the  nearest  numbers  limited  to  six  decimal 
places.  Accordingly,  when  the  fraction  which  is  omitted  ex- 
ceeds half  a  unit  in  the  sixth  decimal  place,  the  last  rigur« 
must  be  increased  by  unity. 

Required  the  logarithm  of  8765432. 
.      The  logarithm  of  8765000  is  6.942752 

Correction  for  the  fifth  figure,  4,  20 

"  "       sixth  figure,  3,  1.5 

"  "       seventh  figure,  2,  0.1 

Therefore  the  logarithm  of  8765432  is  6.942774. 

Required  the  logarithm  of  234567. 

The  logarithm  of  234500  is  O.370143 

Correction  for  the  fifth  figure,  6,  111 

"  "       sixth  figure,  7,  13 

Therefore  the  logarithm  of  234567  is  5.370267. 

To  find  the  Logarithm  of  a  Decimal  Fraction 

(8.)  According  to  Art.  4,  the  decimal  part  of  the  logarithm 
of  any  number  is  the  same  as  that  of  the  number  multiplied 
or  divided  by  10,  100,  1000,  &c.  Hence,  for  a  decimal  frac 


LOGARITHMS.  13 

tion,  we  find  the  logarithm  as  if  the  figures  were  integers,  and 
prefix  the  characteristic  according  to  the  rule  of  Art.  2. 

EXAMPLES. 

The  logarithm  of  345.6  is  2.538574  ; 

"  «  87.65          is  1.942752  : 

4i  2.345        is  0.370143; 

"  .1234      is  1.091315 ; 

«  .005678  is  3.754195. 

To  find  the  Logarithm  of  a  Vulgar  Fraction. 
(9.)  We  may  reduce  the  vulgar  fraction  to  a  decimal,  and 
find  its  logarithm  by  the  preceding  article  ;  or,  since  the  value 
of  a  fraction  is  equal  to  the  quotient  of  the  numerator  divided 
by  the  denominator,  we  may,  according  to  Art.  3,  subtract  the 
logarithm  of  the  denominator  from  that  of  the  numerator ; 
the  difference  will  be  the  logarithm  of  the  fraction. 
Ex.  1.  Find  the  logarithm  of  T3¥,  or  0.1875. 

From  the  logarithm  of  3,  0.477121, 

Take  the  logarithm  of  16,  1.204120. 

Leaves  the  logarithm  of  ft,  or  .1875,  1.273001. 
Ex.  2.  The  logarithm  of  &  is  2.861697. 
Ex.  3.  The  logarithm  of  iff  is  1.147401. 

To  find  the  Natural  Number  corresponding  to  any  Logarithm. 
(10.)  Look  in  the  table,  in  the  column  headed  0,  for  the  first 
two  figures  of  the  logarithm,  neglecting  the  characteristic  ;  the 
other  four  figures  are  to  be  looked  for  in  the  same  column,  01 
in  one  of  the  nine  following  columns  ;  and  if  they  are  exactly 
found,  the  first  three  figures  of  the  corresponding  number  will 
be  found  opposite  to  them  in  the  column  headed  N.,  and  the 
fourth  figure  will  be  found  at  the  top  of  the  page.  This  number 
must  be  made  to  correspond  with  the  characteristic  of  the  given 
logarithm  by  pointing  off  decimals  or  annexing  ciphers.  Thus 
the  natural  number  belonging  to  the  log.  4.370143  is  23450; 

"  "  "  "          1.538574  is  34.56. 

• 

If  the  decimal  part  of  the  logarithm  can  not  be  exactly  found 
in  the  table,  look  for  the  nearest  less  logarithm,  and  take  oui 


14  TRIGONOMETRY 

the  four  figures  of  the  corresponding  natural  number  as  be- 
fore ;  the  additional  figures  may  be  obtained  by  means  of  tho 
Proportional  Parts  at  the  bottom  of  the  page. 

Required  the  number  belonging  to  the  logarithm  4.368399. 

On  page  6,  we  find  the  next  less  logarithm  .368287. 

The  four  corresponding  figures  of  the  natural  number  are 
2335.  Their  logarithm  is  less  than  the  one  proposed  by  112. 
The  tabular  difference  is  186  ;  and,  by  referring  to  the  bottom 
of  page  6,  we  find  that,  with  a  difference  of  186,  the  figure 
corresponding  to  the  proportional  part  112  is  6.  Hence  the 
five  figures  of  the  natural  number  are  23356 ;  and,  since  the 
characteristic  of  the  proposed  logarithm  is  4,  these  five  figures 
are  all  integral. 

Required  the  number  belonging  to  logarithm  5.345678. 

The  next  less  logarithm  in  the  table  is  345570, 

Their  difference  is  108. 

The  first  four  figures  of  the  natural  number  are     2216. 

"With  the  tabular  difference  196,  the  fifth  figure,  correspond, 
ing  to  108,  is  seen  to  be  5,  with  a  remainder  of  10.  To  find 
the  sixth  figure  corresponding  to  this  remainder  10,  we  may 
multiply  it  by  10,  making  100,  and  search  for  100  in  the  same 
line  of  proportional  parts.  We  see  that  a  difference  of  100 
would  give  us  5  in  the  fifth  place  of  the  natural  number. 
Therefore,  a  difference  of  10  must  give  us  5  in  the  sixth  place 
of  the  natural  number.  Hence  the  required  number  is  221655 

In  tho  same  manner  we  find 

'ho  number  corresponding  to  log.  3.538672  is  3456.78 ; 
"  "  "       1.994605  is      98.7654; 

"  "  "       1.647817  is          .444444 

MULTIPLICATION  BY  LOGARITHMS. 

(11.)  According  to  Art.  3,  the  logarithm  of  the  product  of 
two  or  more  factors  is  equal  to  the  sum  of  the  logarithms  of 
those  factors.  Hence,  for  multiplication  by  logarithms,  we 
have  the  following 

RULE. 

Add  the  logarithms  of  the  factors  ;  the  sum  will  be  the  leg 
Gfithm  of  their  product. 

Ex.  1.  Required  the  product  of  57.98  by  18. 


LOGARITHMS.  13 

The  logarithm  of  57.98    -      is  1.763278 
"  «  18  is  1.255273 

The  logarithm  of  the  product  1043.64  is  3.018551 
Ex.  2.  Required  the  product  of  397.65  by  43.78. 

Ans.,  17409.117. 

Ex.  3.  Required  the  continued  product  of  54.32,  6.543,  and 
12.345. 

The  word  sum,  in  the  preceding  rule,  "is  to  be  understood  in 
its  algebraic  sense  ;  therefore,  if  any  of  the  characteristics  of 
the  logarithms  are  negative,  we  must  take  the  difference  be- 
tween their  sum  and  that  of  the  positive  characteristics,  and 
prefix  the  sign  of  the  greater.  It  should  be  remembered  that 
the  decimal  part  of  the  logarithm  is  invariably  positive ;  hence 
that  which  is  carried  from  the  decimal  part  to  the  character- 
istic must  be  considered  positive. 
Ex.  4.  Multiply  0.00563  bjr  17. 

The  logarithm  of  0.00563  is  3.750508 
"  "  17      is  1.230449 

Product,  0.09571,  whose  logarithm  is  2.980957. 

Ex.  5.  Multiply  0.3854  by  0.0576.  Ans.,  0.022199. 

Ex.  6.  Multiply  0.007853  by  0.00476. 

Ans.,  0.0000373S. 

Ex.  7.  Find  the  continued  product  of  11.35,  0.072,  and  0.017. 
(12.)  Negative  quantities  may  be  multiplied  by  means  of 
logarithms  in  the  same  manner  as  positive,  the  proper  sign 
being  prefixed  to  the  result  according  to  the  rules  of  Algebra. 
To  distinguish  the  negative  sign  of  a  natural  number  from  the 
negative  characteristic  of  a  logarithm,  we  append  the  letter  n 
to  the  logarithm  of  a  negative  factor.  Thus 

the  logarithm  of  —56  we  write  1.748188  n. 
Ex.  8.  Multiply  53.46  by  -29.47. 

The  logarithm  of     53.46  is  1.728029 
"  "  -29.47  is  1.469380  n. 

Product,  -1575.47,  log.  3.197409  n. 

Ex.  9.  Find  the  continued  product  of  372.1,  -.0054,  and 
-175.6. 

Ex.  10.  Find  the  continued  product  of  -0.137,  -7.689,  and 
-  0376. 


IB  TRIGONOMETRY. 

DIVISION  BY  LOGARITHMS 

(13.)  According  to  Art.  3,  the  logarithm  of  the  quotient  oi 
one  number  divided  by  another  is  equal  to  the  difference  of 
the  logarithms  of  those  numbers.  Hence,  for  division  by  log- 
arithms,  we  have  the  following 

RULE. 

From  the  logarithm  of  the  dividend,  subtract  the  logarithm 
of  the  divisor  ;  the  difference  will  be  the  logarithm  of  the 
quotient. 

Ex.  1.  Required  the  quotient  of  888.7  divided  by  42.24. 
The  logarithm  of  888.7  is  2.948755 
"  "  42.24  is  1.625724 

The  quotient  is  21.039,  whose  log.  is  1.323031. 
Ex.  2.  Required  the  quotient  of  3807.6  divided  by  13.7. 

Ans.,  277.927. 

The  word  difference,  in  the  preceding  rule,  is  to  be  under- 
etood  in  its  algebraic  sense ;  therefore,  if  the  characteristic  of 
one  of  the  logarithms  is  negative,  or  the  lower  one  is  greater 
than  the  upper,  we  must  change  the  sign  of  the  subtrahend, 
and  proceed  as  in  addition.     If  unity  is  carried  from  the  deci- 
mal part,  this  must  be  considered  as  positive,  and  must  b^ 
united  with  the  characteristic  before  its  sign  is  changed. 
Ex.  3.  Required  the  quotient  of  56.4  divided  l-y  0.00015. 
The  logarithm  of        56.4  is  1.751279 
"  "  0.00015  is  4.176091 

The  quotient  is  376000,  whose  log.  is  5.575188. 
This  result  may  be  verified  in  the  same  way  as  subtraction 
hi  common  arithmetic.  The  remainder,  added  to  the  subtra- 
hend, should  be  equal  to  the  minuend.  This  precaution  should 
always  be  observed  when  there  is  any  doubt  with  regard  to 
the  sign  of  the  result. 

Ex.  4.  Required  the  quotient  of  .8692  divided  by  42.258. 

Ans. 

Ex.  5.  Required  the  quotient  of  .74274  divided  by  .00928. 
Ex.  6.  Required  the  quotient  of  24,934  divided  by  .078541 
Negative  quantities  may  be  divided  by  means  of  logarithms 


LOGARITHMS.  17 

in  the  same  manner  as  positive,  the  pr  >per  sign  being  prefixed 
to  the  result  according  to  the  rules  of  Algebra. 

Ex.  7.  Required  the  quotient  of  -79.54  divided  by  0  08321 
Ex.  8.  Required  the  quotient  of  -0.4753  divided  by  -36.74. 

INVOLUTION  BY  LOGARITHMS. 

(14.)  It  is  proved  in  Algebra,  Art.  340,  that  the  logarithm 
of  any  power  of  a  number  is  equal  to  the  logarithm  of  that 
number  multiplied  by  the  exponent  of  the  power.  Hence,  to 
involve  a  number  by  logarithms,  we  have  the  folk  wing 

RULE. 

Multiply  the  logarithm  of  the  number  by  the  exponent  of 
the  power  required. 

Ex.  1.  Required  the  square  of  428. 

The  logarithm  of  428  is  2.631444 


Square,  183184,  log.  5.262888. 
Ex.  2.  Required  the  20th  power  of  1.06. 

The  logarithm  of  1.06  is  0.025306 

20 

20th  power,  3.2071,  log.  0.506120. 

Ex.  3.  Required  the  5th  power  of  2.846. 

It  should  be  remembered,  that  what  is  carried  from  the  dec- 
imal part  of  the  logarithm  is  positive,  whether  the  characteris- 
tic is  positive  or  negative. 

Ex.  4,  Required  the  cube  of  .07654. 

The  logarithm  of  .07654  is  2.883888 

3 

Cube,  .0004484,  log.  1.651664. 
Ex.  5.  Required  the  fourth  power  of  0.09874. 
Ex.  6.  Required  the  seventh  power  of  0.8952. 

EVOLUTION  BY  LOGARITHMS. 

(15.)  It  is  proved  in  Algebra,  Art.  341,  that  the  logarithm 
jl  any  root  of  a  number  is  equal  to  the  logarithm  of  that  num- 
oer  divided  by  the  index  of  the  root.     Hence,  to  extract  the 
oot  of  a  number  by  logarithms,  we  have  the  following 

B 


18  TRIGOKOMETRY. 

RULE. 

Divide  the  logarithm  of  the  number  by  the  index  of  iht 
root  required. 

Ex.  1.  Required  the  cube  root  of  482.38. 

The  logarithm  of  482.38  is  2.683389. 

Dividing  by  3,  we  have  0.894463,  which  corresponds  to 
7.842,  which  is  therefore  the  root  required. 

Ex.  2.  Required  the  100th  root  of  365. 

Am.,  1.0608. 

When  the  characteristic  of  the  logarithm  is  negative,  and  is 
not  divisible  by  the  given  divisor,  we  may  increase  the  char- 
acteristic by  any  number  which  will  make  it  exactly  divisible, 
provided  we  prefix  an  equal  positive  number  to  the  decimal 
part  of  the  logarithm. 

Ex.  3.  Required  the  seventh  root  of  0.005846. 

The  logarithm  of  0.005846  is  3.766859,  which  may  be  writ- 
ten 7+4.766859. 

Dividing  by  7,  we  have  1.680980,  which  is  the  logarithm  of 
4797,  which  is,  therefore,  the  root  required. 

This  result  may  be  verified  by  multiplying  1.680980  by  7, 
Mie  result  will  be  found  to  be  3.766860. 

Ex.  4.  Required  the  fifth  root  of  0.08452. 

Ex.  5.  Required  the  tenth  root  of  0.007815. 

PROPORTION  BY  LOGARITHMS. 

(16.)   The  fourth  term  of  a  proportion  is  found  by  multiply 
ing  together  the  second  and  third  terms,  and  dividing  by  the 
first.     Hence,  to  find  the  fourth  term  of  a  proportion  by  loga 
rithms, 

Add  the  logarithms  of  the  second  and  third  terms,  and  front 
their  sum  subtract  the  logarithm  of  the  first  term. 

Ex.  1.  Find  a  fourth  proportional  to  72.34, 2.519,  and  357.4S 

Ans.,  12,448. 

(17.)  When  one  logarithm  is  to  be  subtracted  from  anothej, 
it  may  be  more  convenient  to  convert  the  subtraction  into  an 
addition,  which  may  be  done  by  first  subtracting  the  given  log. 
arithm  from  10,  adding  the  difference  to  the  other  logarithm 
and  afterward  rejecting  the  10. 


LOGARITHMS.  19 

The  difference  between  a  given  logarithm  ana  10  is  called 
its  complement ;  and  this  is  easily  taken  from  the  table  by  be- 
ginning at  the  left  hand,  subtracting  each  figure  from  9,  ex- 
cept  the  last  significant  figure  on  the  right,  which  must  be 
subtracted  from  10. 

To  subtract  one  logarithm  from  another  is  the  same  as  to 
add  its  complement,  and  then  reject  10  from  the  result.  For 
a—b  is  equivalent  to  10— b+a— 10. 

To  work  a  proportion,  then,  by  logarithms,  we  must 
Add  the  'complement  of  the  logarithm  of  the  first  term  to 
the  logarithms  of  the  second  and  third  terms. 

The  characteristic  must  afterward  be  diminished  by  10. 
Ex.  2.  Find  a  fourth  proportional  to  6853,  489,  and  38750. 
The  complement  of  the  logarithm  of  6853  is  6.164119 
The  logarithm  of  489  is  2.689309 

"  "  38750  is  4.588272 

The  fourth  term  is  2765,  whose  logarithm  is  3.441700. 
One  advantage  of  using  the  complement  of  the  first  term  in 
working  a  proportion  by  logarithms  ig>  that  it  enables  us  to 
exhibit  the  operation  in  a  more  compact  form. 

Ex.  3.  Find  a  fourth  proportional  to  73.84,  658.3,  and  4872. 

Ans. 
Ex.  4.  Find  a  fcurth  proportional  to  5.745,  781.2,  and  54.2"* 


BOOK  II, 


PLANE  TRIGONOMETRY, 

(18.)  TRIGONOMETRY  is  the  science  which  leaches  how  to  de- 
termine the  several  parts  of  a  triangle  from  having  certain 
parts  given. 

Plane  Trigonometry  treats  of  plane  triangles ;  Spherical 
Trigonometry  treats  of  spherical  triangles. 

(19.)  The  circumference  of  every  circle  is  supposed  to  be 
divided  into  360  equal  parts,  called  degrees  ;  each  degree  into 
60  minutes,  and  each  minute  into  60  seconds.  Degrees,  min- 
utes, and  seconds  are  designated  by  the  characters  °,  ',  ". 
Thus  23°  14'  35"  is  read  23  degrees,  14  minutes,  and  35  sec- 
onds. 

Since  an  angle  at  the  center  of  a  circle  is  measured  by  the 
arc  intercepted  by  its  sides,  a  right  angle  is  measured  by  90°, 
two  right  angles  by  180°,  and  four  right  angles  are  measured 
by  360°. 

(20.)  The  complement  of  an  arc  is  what  remains  after  sub- 
tracting  the  arc  from  90°  •     Thus  the 
*rc  DF  is   the   complement  of  AF. 
The  complement  of  25°  15'  is  64°  45'. 

In  general,  if  we  represent  any  arc 
by  A,  its  complement  is  90°— A. 
Hence,  if  an  arc  is  greater  than  90°, 
its  complement  must  be  negative. 
Thus,  the  complement  of  100°  15'  is 
—  10°  15'.  Since  the  two  acute  an- 
gles of  a  right-angled  triangle  are  to- 
gether equal  to  a  right  angle,  each  of  them  must  be  the  com- 
plement of  the  other. 

(21.)  The  supplement  of  an  arc  is  \rhai  remains  after  sub- 
cracting  the  arc  from  180°.  Thus  the  arc  BDF  is  the  supple- 
ment of  the  arc  AF.  The  supplement  of  25°  15'  is  154°  45'. 

In  general,  if  we  represent  any  arc  by  A,  its  supplement  is 


PLANE    TRIGONOMETRY.  21 

180° —A.  Hence,  if  an  arc  is  greater  than  180%  its  supple- 
ment must  be  negative.  Thus  the  supplement  of  200°  is  —20° 
Since  in  every  triangle  the  sum  of  the  three  angles  is  180°. 
either  angle  is  the  supplement  of  the  sum  of  the  other  two. 

(22.)  The  sine  of  an  arc  is  the  perpendicular  let  fall  from 
one  extremity  of  the  arc  on  the  radius  passing  through  the 
other  extremity.  Thus  FGr  is  the  sine  of  the  arc  AF,  or  of  the 
angle  ACF. 

Every  sine  is  half  the  chord  of  double  the  arc.  Thus  the 
sine  FG-  is  the  half  of  FH,  which  is  the  chord  of  the  arc  FAH, 
double  of  FA.  The  chord  which  subtends  the  sixth  part  of 
the  circumference,  or  the  chord  of  60°,  is  equal  to  the  radius 
(Geom.j  Prop.  IV.,  B.  VI.) ;  hence  the  sine  of  30°  is  equal  to 
half  of  the  radius. 

(23.)  The  versed  sine  of  an  arc  is.  that  part  of  the  diameter 
intercepted  between  the  sine  and  the  arc.  Thus  GrA  is  the 
versed  sine  of  the  arc  AF. 

(24).  The  tangent  of  an  arc  is  the  line  which  touches  it  at 
one  extremity,  and  is  terminated  by  a  line  drawn  from  the 
center  through  the  other  extremity.  Thus  AI  is  the  tangent 
of  the  arc  AF,  or  of  the  angle  ACF. 

(25.)  The  secant  of  an  arc  is  the  line  drawn  from  the  cen- 
ter of  the  circle  through  one  extremity  of  the  arc,  and  is  lim 
ited  by  the  tangent  drawn  through  the  other  extremity. 

Thus  CI  is  the  secant  of  the  arc  AF,  or  of  the  angle  ACF. 

(26.)  The  cosine  of  an  arc  is  the  sine  of  the  complement  of 
that  arc.  Thus  the  arc  DF,  being  the  complement  of  AF,  FK 
is  the  sine  of  the  arc  DF,  or  the  cosine  of  the  arc  AF. 

The  cotangent  of  an  arc  is  the  tangent  of  the  complement 
of  that  arc.  Thus  DL  is  the  tangent  of  the  arc  DF,  or  the  co- 
tangent of  the  arc  AF. 

The  cosecant  of  an  arc  is  the  secant  of  the  complement  of 
that  arc.  Thus  CL  is  the  secant  of  the  arc  DF,  or  the  cose- 
cant  of  the  arc  AF. 

In  genera],  if  we  repr'  ent  any  angle  by  A, 
cos.      A=sine    (90°-A). 
cot.      A=tang.  (90°-A). 
cosec.  A=sec.     (90°— A). 

Since,  in  a  right-angled  triangle,  either  of  the  acute  an<ne* 


T  H  I  G  0  N  0  M  E  T  R  Y. 


is  the  complement  of  the  other,  the  sine,  tangent,  and  secant 
<>f  one  of  these  angles  is  the  cosine,  cotangent,  and  cosecan 
of  the  other. 

(27.)  The  sine,  tangent,  and  secant  of  an  arc  are  equal  to 
the  sine,  tangent,  and  secant  of  its  supplement.  Thus  FG  h 
the  sine  of  the  arc  AF,  or  of  its  sup- 
plement, BDF.  Also,  AI,  the  tan- 
gent of  the  arc  AF,  is  equal  to  BM, 
the  tangent  of  the  arc  BDF.  And 
CI,  the  secant  of  the  arc  AF,  is  equal 
to  CM,  the  secant  of  the  arc  BDF. 

The  versed  sine  of  an  acute  angle, 
ACF,  is  equal  to  the  radius  minus 
the  cosine  CG.  The  versed  sine  of 
an  obtuse  angle,  BCF,  is  equal  to  ra- 
dius plus  the  cosine  CGr ;  that  is,  to  BG. 

(28.)  The  relations  of  the  sine,  cosine,  &c.,  to  each  other, 
may  be  derived  from  the  proportions  of  the  sides  of  similar 
triangles.  Thus  the  triangles  CG-F,  CAI,  CDL,  being  similar, 
we  have, 

1.  CG  :  GF  :  :  CA  :  AI ;  that  is,  representing  the  arc  by  A, 
and  the  radius  of  the  circle  by  R,  cos.  A  :  sin.  A  :  :  R  :  tang.  A. 

R  sin.  A 
Whence  tang.  A= 


2.  CG  :  :  CF  :  CA 


CI;  that  is,  cos.  A  :  R  : 

Whence  sec.  A= 


cos.  A' 
3   GF  :  CG  : :  CD  :  DL ;  that  is,  sin.  A  :  cos.  A  :  :  R  :  cot.  A 

R  cos.  A 

Whence  cot.  A= — : — - — . 
sin.  A 

that  is,  sin.  A  :  R  : :  R  :  cosec.  A. 

•pa 

Whence  cosec.  A=— — '-r-. 
sin.  A 

CD  :  DL ;  that  is,  tang.  A  :  R  : :  R  :  cot.  A. 

Rs 

Whence  tang.  A.= r-- 

cot.  A 

The  preceding  values  of  tangent  and  cotangent,  secant  and 
cosecant  will  be  frequently  referred  to  hereafter,  and  should 
be  carefully  committed  to  memory. 


4    GF:CF::CD:CL 


5.  AI :  AC 


PLANE    TRIGONOMETRY. 


Also,  in  the  right-angled  triangle  CGF,  we  find 
UFa;  that  is,  sin.  2A+cos.  2A=R2  ;  or, 

The  square  of  the  sine  of  an  arc,  together  with  the  square 
tf  its  cosine,  is  equal  to  the  square  of  the  radius.     - 

Hence  sin.  A=±  VR2—  cos.  2A. 
And  cos.     A= 


—  sin.  2A. 

(29.)  A  table  of  natural  sines,  tangents,  &c.,  is  a  table  giv- 
ing the  lengths  of  those  lines  for  different  angles  in  a  circle 
whose  radius  is  unity. 

Thus,  if  we  describe  a  circle  with  a  radius  of  one  inch,  and 
divide  the  circumference  into  equal  parts  of  ten  degrees,  we 
shall  find 

the  sine  of  10°  equals  0.174  inch  ; 


"       "       20°      "      0.342     "            g 
"       "       30°      "      0.500     " 
«       "      40°      "      0.643     " 
"       "       50°      "      0.766     " 
"       "       60°      "      0.866     " 
«       «       70°      "      0.940     " 
"       "       80°      "      0.985     " 
"       «       90°      "      1.000     " 

°'   8 

| 

9* 

5o° 
\ 

50° 
\ 

40 
\ 

\ 

30° 
V 

\ 

If  we  draw  the  tangents  of  the  same  arcs,  we  shall  find 
the  tangent  of  10°  equals  0.176  inch ; 


20° 
30° 
40° 
45° 
50° 
60° 
70° 
80° 
90° 


0.364 
0.577 
0.839 
1.000 
1.192 
1.732 
2.747 
5.671 
infinite. 


Also,  if  we  draw  the  secants  of  the  same 
arcs,  we  shall  find  that 

the  secant  of  10°  equals  1.015  inch ; 
"  "         20°      "      1.064     " 

"          "         30°      "      1.155     " 
u          u         40°      «      1.305     " 


2^  TRIGONOMETRY. 

the  secant  of  50°  equals  1.556  incli ; 
"         "         60°      "      2.000     " 
«         u         7Q3      «      2.924     « 

«•         "         80°      "      5.759     " 
"          "         90°       "      infinite. 

In  the  accompanying  table,  pages  116—133,  the  sines,  co« 
sines,  tangents,  and  cotangents  are  given  for  every  minute  of 
the  quadrant  to  six  places  of  figures. 

(30.)  To  find  from  the  table  the  natural  sine,  cosine,  fyc., 
of  an  arc  or  angle. 

If  a  sine  is  required,  look  for  the  degrees  at  the  top  of  the 
page,  and  for  the  minutes  on  the  left ;  then,  directly  under  the 
given  number  of  degrees  at  the  top  of  the  page,  and  opposite 
to  the  minutes  on  the  left,  will  be  found  the  sine  required. 
Since  the,  radius  of  the  circle  is  supposed  to  be  unity,  the  sine 
of  every  arc  below  90°  is  less  than  unity.  The  sines  are  ex- 
pressed in  decimal  parts  of  radius  ;  and,  although  the  decimal 
point  is  not  written  in  the  table,  it  must  always  be  prefixed. 
As  the  first  two  figures  remain  the  same  for  a  great  many 
numbers  in  the  table,  they  are  only  inserted  for  every  ten  min- 
utes, and  the  vacant  places  must  be  supplied  from  the  two 
leading  figures  next  preceding  Thus,  on 

page  130,  the  sine  of  25°  11'  is  0.425516  ; 
page  126,    «        "       51°  34'  is  0.783332,  &o. 
The  tangents  are  found  in  a  similar  manner.     Thus 
the  tangent  of  31°  44'  is  0.618417  ; 

"  "         65°  27'  is  2.18923. 

The  same  number  in  the  table  is  both  the  sine  of  an  arc  arid 
the  cosine  of  its  complement.  The  degrees  for  the  cosines 
must  be  sought  at  the  bottom  of  the  page,  and  the  minutes  on 
the  right.  Thus, 

on  page  130,  the  cosine  of  16°  42'  is  0.957822 ; 
on  page  118,    "         "         73°  17'  is  0.287639,  &o. 
The  cotangents  are  found  in  the  same  manner.     Thus 
the  cotangent  of  19°  16'  is  2.86089 ; 
«  "          54°  53'  is  0.703246. 

It  is  not  necessary  to  extend  the  tables  beyond  a  quadrant, 
because  the  sine  of  an  angle  is  equal  to  that  of  its  supplement 
(Art.  27).  Thus 


PLANE    TRIGONOMETRY.  23 

the  sine  of  143°  24'  is  0.596225  ; 

"  cosine        of  151°  23'  is  0.877844; 
"  tangent     of  132°  36'  is  1.08749  ; 
"    cotangent  of  116°    7'  is  0.490256,  &o. 
(31.)  If  a  sine  is  required  for  an  arc  consisting  of  degrees, 
minutes,  and  seconds,  we  must  make  an  allowance  for  the  sec- 
onds in  the  same  manner  as  was  directed  in  the  case  of  loga- 
rithms, Art.  7  ;  for,  within  certain  limits,  the  differences  of  the 
sines  are  proportional  to  the  differences  of  the  corresponding 
arcs.     Thus 

the  sine  of  34°  25'  is  .565207  ; 
«       «       340  26'  is  .565447. 

The  difference  of  the  sines  corresponding  to  one  minute  oi 
arc,  or  60  seconds,  is  .000240.  The  proportional  part  for  1 '  is 
found  by  dividing  the  tabular  difference  by  60,  and  -the  quo- 
tient, .000004,  is  placed  at  the  bottom  of  page  122,  in  the  col- 
amn  headed  34°.  The  correction  for  any  number  of  seconds 
will  be  found  by  multiplying  the  proportional  part  for  V  by 
the  number  of  seconds. 

Required  the  natural  sine  of  34°  25'  37". 
The  proportional  part  for  1",  being  multiplied  by  37,  becomes 
148,  which  is  the  correction  for  37".     Adding  this  to  the  sine 
of  34°  25',  we  find 

the  sine  of  34°  25'  37"  is  .565355. 

Since  the  proportional  part  for  1"  is  given  to  hundredths  of  a 
unit  in  the  sixth  place  of  figures,  after  we  have  multiplied  by 
the  given  number  of  seconds,  we  must  reject  the  last  two  fig- 
ures of  the  product. 

Tn  the  same  mamier  we  find 

the  cosine  of  56°  34'  28"  is  .550853. 

It  will  be  observed,  that  since  the  cosines  decrease  while 
the  arcs  increase,  the  correction  for  the  28"  is  to  be  subtra  cted 
from  the  cosine  of  56°  34'. 
Tn  the  same  manner  we  find 

the  natural  sine  of  27°  17;  12"  is  0.458443  ; 

"         «      cosine        of  45C  23'  23"  is  0.702281 ; 
"         "      tangent     of  63'  32'  34"  is  2.00945  : 
• «         "      cotangent  of  81°  48'  56"  is  0.143825 


26  TRIGONOMETRY. 


To  find  the  i  umber  of  degrees,  minutes^  and  second! 
Idonging  to  a  given  sine  or  tangent. 

If  the  given  sine  or  tangent  is  found  exactly  in  the  tablej 
the  corresponding  degrees  will  be  found  at  the  top  of  the  page, 
and  the  minutes  on  the  left  hand.  But  when  the  given  num- 
ber is  not  found  exactly  in  the  table,  look  for  the  sine  or  tan- 
gent which  is  next  less  than  the  proposed  one,  and  take  out 
the  corresponding  degrees  and  minutes.  Find,  also,  the  dif- 
ference between  this  tabular  number  and  the  number  proposed, 
and  divide  it  by  the  proportional  part  for  1"  found  at  the  bot 
torn  of  the  page  ;  the  quotient  will  be  the  required  number  of 
seconds. 

Required  the  arc  whose  sine  is  .750000. 

The  next  less  sine  in  the  table  is  .749919,  the  arc  correspond- 
ing  to  which  is  48°  35'.  The  difference  between  this  sine  and 
that  proposed  is  81,  which,  divided  by  3.21,  gives  25.  Hence 
the  required  arc  is  48°  35'  25". 

Tn  the  same  manner  we  find 

the  arc  whose  tangent  is  2.00000  is  63°  26'  6". 

If  a  cosine  or  cotangent  is  required,  we  must  look  for  the 
number  in  the  table  which  is  next  greater  than  the  one  pro- 
posed, and  then  proceed  as  for  a  sine  or  tangent.     Thus 
the  arc  whose  cosine        is    .40000  is  66°  25'  18"  ; 
"     "        "      cotangent  is  1.99468  is  26°  37'  34". 

(33.)  On  pages  134-5  will  be  found  a  table  of  natural  se- 
cants for  every  ten  minutes  of  the  quadrant,  carried  to  seven 
places  of  figures.  The  degrees  are  arranged  in  order  in  the 
first  vertical  column  on  the  left,  and  the  minutes  at  the  top 
of  the  page.  Thus 

the  secant  of  21°  20'  is  1.073561  ; 
«         81°  50'  is  7.039622. 

If  a  secant  is  required  for  a  number  of  minutes  not  given  in 
the  table,  the  correction  for  the  odd  minutes  may  be  found  by 
means  of  the  last  vertical  column  on  the  right,  which  shows 
the  proportional  part  for  one  minute. 

Let  it  be  required  to  find  the  secant  of  30°  33' 
The  secant  of  30°  30'  is  1.160592. 

The  correction  for  1'  is  198.9,  which,  multiplied  by  3,  bo- 


PLANE    TR  IGONOMETIIY.  27 

somes  597.     Adding  this  to  the  number  before  found,  we  ob- 
tain  1.161189. 

For  a  cosecant,  the  degrees  must  be  sought  in  the  right- 
hand  vertical  column,  and  the  minutes  at  the  bottom  of  th* 
page.  Thus 

the  cosecant  of  47°  40'  is  1.352742  ; 
«  "         38°  33'  is  1.604626. 

(34.)  When  the  natural  sines,  tangents,  &c.,  are  used  in  pro- 
portions, it  is  necessary  to  perform  the  tedious  operations  of 
multiplication  and  division.  It  is,  therefore,  generally  prefer- 
able to  employ  the  logarithms  of  the  sines ;  and,  for  conven- 
ience, these  numbers  are  arranged  in  a  separate  table,  called 
logarithmic  sines,  &c.  Thus 

the  natural  sine  of  14°  30'  is  0.250380. 

Its  logarithm,  found  from  page  6,  is  1.398600. 

The  characteristic  of  the  logarithm  is  negative,  as  must  be 
the  case  with  all  the  sines,  since  they  are  less  than  unity.  To 
avoid  the  introduction  of  negative  numbers  in  the  table,  we  in- 
crease the  characteristic  "by  10.  making  9.398600,  and  this  is 
the  number  found  on  page  38  for  the  logarithmic  sine  of  14° 
30'.  The  radius  of  the  table  of  logarithmic  sines  is  therefore, 
properly,  10,000,000,000,  whose  logarithm  is  10. 

(35.)  The  accompanying  table  contains  the  logarithmic  sines 
and  tangents  for  every  ten  seconds  of  the  quadrant.  The  de- 
grees and  seconds  are  placed  at  the  top  of  the  page,  and  the 
minutes  in  the  left  vertical  column.  After  the  first  two  de- 
grees, the  three  leading  figures  in  the  table  of  sines  are  only 
given  in  the  column  headed  0",  and  are  to  be  prefixed  to  the 
numbers  in  the  other  columns,  as  in  the  table  of  logarithms  of 
numbers.  Also,  where  the  leading  figures  change,  this  change 
is  indicated  by  dots,  as  in  the  former  table.  The  correction 
for  any  number  of  seconds  less  than  10  is  given  at  the  bottom 
of  the  page. 

(36.)  To  find  the  logarithmic  sine  or  tangent  of  a  given 
arc. 

Look  for  the  degrees  at  the  top  of  the  page,  the  minutes  on 
the  left  hand,  and  the  next  less  number  of  seconds  at  the  top ; 
then,  under  the  seconds,  and  opposite  to  the  minutes,  will  be 
tound  four  figures,  to  which  the  three  leading  figures  are  to  be 


38  TRIGONOMETRY, 

prefixed  from  the  column  headed  Ox/ ;  to  this  add  the  proper* 
tional  part  for  the  odd  seconds  at  the  bottom  of  the  page. 
Required  the  logarithmic  sine  of  24°  27'  34". 

The  logarithmic  sine  of  24°  27'  30"  is  9.617033 
Proportional  part  for  4"  is  18 

Logarithmic  sine  of  24°  27'  34"  is       9.617051. 
Required  the  logarithmic  tangent  of  73°  35'  43". 
The  logarithmic  tangent  73°  35'  40"  is  10.531031 
Proportional  part  for  3"  is  23 

Logarithmic  tangent  of  73°  35'  43"  is    10.531054. 
When  a  cosine  is  required,  the  degrees  and  seconds  must  be 
sought  at  the  bottom  of  the  page,  and  the  minutes  on  the  right, 
and  the  correction  for  the  odd  seconds  must  be  subtracted  from 
the  number  in  the  table. 

Required  the  logarithmic  cosine  of  59°  33'  47". 

The  logarithmic  cosine  of  59°  33'  40"  is  9.704682 
Proportional  part  for  7"  is  25 

Logarithmic  cosine  of  59°  33'*47"  is        9.704657. 
So,  also,  the  logarithmic  cotangent  of  37°  27'  14"  is  found 
to  be  10.115744. 

It  will  be  observed  that  for  the  cosines  and  cotangents,  the 
seconds  are  numbered  from  10"  to  60",  so  that  if  it  is  re- 
quired to  find  the  cosine  of  25°  25'  0"  we  must  look  for  25° 
24'  60" ;  and  so,  also,  for  the  cotangents. 

(37.)  The  proportional  parts  given  at  the  bottom  of  each 
page  correspond  to  the  degrees  at  the  top  of  the  page,  in- 
creased by  30',  and  are  not  strictly  applicable  to  any  other 
number  of  minutes ;  nevertheless,  the  differences  of  the  sin^s 
change  so  slowly,  except  near  the  commencement  of  the  quad- 
rant, that  the  error  resulting  from  using  these  numbers  for 
every  part  of  the  page  will  seldom  exceed  a  unit  in  the  sixth 
decimal  place.  For  the  first  two  degrees,  the  differences 
change  so  rapidly  that  the  proportional  part  for  1"  is  given  for 
each  minute  in  the  right-hand  column  of  the  page.  The  cor- 
rection  for  any  number  of  seconds  less  than  ten  will  be  fouru 
by  multiplying  the  proportional  part  for  1"  by  the  given  num. 
ber  of  seconds. 

Required  the  logarithmic  sine  of  1°  17'  33", 


PLANS    TRIGONOMETRY.  2Jv 

The  logarithmic  sine  of  1°  17'  30"  is  8.352991. 

The  correction  for  3"  is  found  by  multiplying  93.4  by  3 
which  gives  280.  Adding  this  to  the  above  tabular  number, 
we  obtain  for 

the  sine  of  1°  17'  33",  8.353271. 

A  similar  method  may  be  employed  for  several  of  the  first 
degrees  of  the  quadrant,  if  the  proportional  parts  at  the  bottom 
of  the  page  are  not  thought  sufficiently  precise.  This  correc- 
tion may,  however,  be  obtained  pretty  nearly  by  inspection, 
from  comparing  the  proportional  parts  for  two  successive  de- 
grees. Thus,  on  page  26,  the  correction  for  1",  corresponding 
to  the  sine  of  2°  30',  is  48 ;  the  correction  for  1",  correspond- 
ing to  the  sine  of  3°  30',  is  34.  Hence  the  correction  for  1", 
corresponding  to  the  sine  of  3°  0',  must  be  about  41 ;  and,  in 
the  same  manner,  we  may  proceed  for  any  other  part  of  the 
table. 

(38.)  Near  the  close  of  the  quadrant,  the  tangents  vary  so 
rapidly  that  the  same  arrangement  of  the  table  is  adopted  as 
for  the  commencement  of  the  quadrant.  For  the  last,  as  well 
as  the  first  two  degrees  of  the  quadrant,  the  proportional  part 
to  1"  is  given  for  each  minute  separately.  These  proportional 
parts  are  computed  for  the  minutes  placed  opposite  to  them, 
increased  by  30",  and  are  not  strictly  applicable  to  any  other 
number  of  seconds ;  nevertheless,  the  differences  for  the  most 
part  change  so  slowly,  that  the  error  resulting  from  using  these 
numbers  for  every  part  of  the  same  horizontal  line  is  quite 
small.  "When  great  accuracy  is  required,  the  table  on  page  114 
may  be  employed  for  arcs  near  the  limits  of  the  quadrant.  This 
table  furnishes  the  differences  between  the  logarithmic  sinea 
and  the  logarithms  of  the  arcs  expressed  in  seconds.  Thus 

the  logarithmic  sine  of  0°  5',  from  page  22,  is  7.162696 

the  logarithm  of  300"  (-5')  is  2.477121 

the  difference  is  4.685575. 

This  is  the  number  found  on  page  114,  under  the  heading 
log-,  sine  A— log'.  A",  opposite  to  5  min. ;  and,  in  a  similar  man- 
ner, the  other  numbers  in  the  same  column  are  obtained.  These 
numbers  vary  quite  slowly  for  two  degrees  ;  and  henc°,  to  find 
the  logarithmic  sine  of  an  arc  less  than  two  degrees  we  ha  T<* 


30  TRIGONOMETRY. 

but  to  add  the  logarithm  of  the  arc  expressed  in  seconds  to  thfl 
appropriate  number  found  in  this  table. 
Required  the  logarithmic  sine  of  0°  7'  22". 

Tabular  number  from  page  114,  4.685575 
The  logarithm  of  442"  is  2.645422 

Logarithmic  sine  of  0°  7'  22"  is  7.330997. 
The  logarithmic  tangent  of  an  arc  less  than  two  degrees  ia 
found  in  a  similar  manner. 

Required  the  logarithmic  tangent  of  0°  27'  36". 

Tabular  number  from  page  114,          4.685584 
The  logarithm  of  1656"  is  3.219060 

Logarithmic  tangent  of  0°  27'  36"  is  7.904644. 
The  column  headed  log.  cot.  A+log.  A",  is  found  by  adding 
'ho  logarithmic  cotangent  to  the  logarithm  of  the  arc  expressed 
in  seconds.  Hence,  to  find  the  logarithmic  cotangent  of  an  aro 
less  than  two  degrees,  we  must  subtract  from  the  tabular  num. 
bsr  the  logarithm  of  the  arc  in  seconds. 

Required  the  logarithmic  cotangent  of  0°  27'  36". 
Tabular  number  from  page  114,  15.314416 

The  logarithm  of  1656"  is  3.219060 

Logarithmic  cotangent  of  0°  27'  36"  is  12.095356. 
The  same  method  will,  of  course ,  furnish  cosines  and  cotan 
gents  of  arcs  near  90°. 

(39.)  The  secants  and  cosecants  are  omitted  in  this  table, 
since  they  are  easily  derived  from  the  cosines  and  sines.  Wc< 

Ra 
have  found,  Art.  28,  secant  = — : — ;  or,  taking  the  logarithms, 

log.  secant  =2.  log.  R— log.  cosine 

=20 —log.  cosine. 

Ra 

Also,  cosecant          =  - — , 

sine 

or  log.  cosecant        =20— log.  sine.     That  is, 

The  logarithmic  secant  is  found  by  subtracting  the  loga- 
rithmic cosine  from  20 ;  and  the  logarithmic  cosecant  is  found 
by  subtracting  the  logarithmic  sine  from  20. 

Thus  we  have  found  the  logarithmic  sine  of  24°  27'  34'  to 
be  9.617051. 

Hence  tho  logarithmic  cosecant  of  24°  27'  34"  is  10.382949 


PLANE    TRIGONOMETRY.  31 

The  logarithmic  cosine  of  54°  12'  40"  is    9.767008. 

Hence  the  logarithmic  secant  of  54°  12'  40"  is  10.232992. 

(40.)  To  find  the  arc  corresponding  to  a  given  logarithmic 
sine  or  tangent. 

If  the  given  number  is  found  exactly  in  the  table,  the  cor- 
responding degrees  and  seconds  will  be  found  at  the  top  of  the 
page,  and  the  minutes  on  the  left.  But  when  the  given  num- 
ber is  not  found  exactly  in  the  table,  look  for  the  sine  or  tan- 
gent which  is  next  less  than  the  proposed  one,  and  take  out 
the  corresponding  degrees,  minutes,  and  seconds.  Find,  also, 
the  difference  between  this  tabular  number  and  the  number 
proposed,  and  corresponding  to  this  difference,  at  the  bottom 
of  the  page,  will  be  found  a  certain  number  of  seconds  which 
is  to  be  added  to  the  arc  before  found. 

Required  the  arc  corresponding  to  the  logarithmic  sine 
9.750000. 

The  next  less  sine  in  the  table  is  9.749987. 

The  arc  corresponding  to  which  is  34°  13'  0". 

The  difference  between  its  sine  and  the  one  proposed  is  13, 
corresponding  to  which,  at  the  bottom  of  the  page,  we  find  4'1 
nearly.  Hence  the  required  arc  is  34°  13'  4". 

In  the  same  manner,  we  find  the  arc  corresponding  to  loga- 
rithmic tangent  10.200000  to  be  60°  38'  57". 

When  the  arc  falls  within  the  first  two  degrees  of  the  quad- 
rant, the  odd  seconds  may  be  found  by  dividing  the  difference 
between  the  tabular  number  and  the  one  proposed,  by  the  pro- 
portional part  for  1".  We  thus  find  the  arc  corresponding  to 
logarithmic  sine  8.400000  to  be  1°  26'  22"  nearly. 

We  may  employ  the  same  method  for  the  last  two  degrees 
of  the  quadrant  when  a  tangent  is  given ;  but  near  the  limits 
of  the  quadrant  it  is  better  to  employ  the  auxiliary  table  on 
page  114.  The  tabular  number  on  page  114  is  equal  to  log. 
sin.  A— log.  A".  Hence  log.  sin.  A— tabular  number  =log. 
A"  ;  that  is,  if  we  subtract  the  corresponding  tabular  number 
on  page  114,  from  the  given  logarithmic  sine,  the  remainder 
will  be  the  logarithm  of  the  arc  expressed  in  seconds. 

Required  the  arc  corresponding  to  logarithmic  sine  7.000000. 

We  see,  from  page  22,  that  the  arc  must  be  nearly  3' ;  thff 
corresponding  tabular  number  on  page  114  is  4.685575 


1)2  T  R  I  G  O  N  O  M  E  T  R  Y. 

The  difference  is  2.314425. 
which  is  the  logarithm  of  206."265. 

Hence  the  required  arc  is  3'  .26. "265. 

Required  the  arc  corresponding  to  log.  sine  8.000000. 

"We  see  from  page  22,  that  the  arc  is  about  34'.  The  cor- 
responding tabular  number  from  page  114  is  4.685568,  which,, 
subtracted  from  8.000000,  leaves  3.314432,  which  is  the  log- 
arithm of  2062. "68.  Hence  the  required  arc  is 

34'  22."68. 

In  the  same  manner,  we  find  the  arc  corresponding  to  loga 
rithmic  tangent  8.184608  to  be  0°  52'  35". 

SOLUTIONS  OF  RIGHT-ANGLED  TRIANGLES. 
THEOREM  I. 

(41.)  In  any  right-angled  triangle,  radius  is  to  the  hypoth- 
en?tse  as  the  sine  of  either  acute  angle  is  to  the  opposite 
or  the  cosine  of  either  acute  angle  to  the  adjacent  side. 

Let  the  triangle  CAB  be  right  angled 
at  A,  then  will 

R  :  CB  :  :  sin.  C  :  BA  : :  cos.  C  :  CA. 

From  the  point  C  as  a  center,  with  a 
radius  equal  to  the  radius  of  the  tables,   C  E  D      -A- 

describe  the  arc  DE,  and  on  AC  let  fall  the  perpendicular  EF 
Then  EF  will  be  the  sine,  and  CF  the  cosine  of  the  angle  C. 
Because  the  triangles  CAB,  CFE  are  similar,  we  have 

CE  :  CB  :  :  EF  :  BA, 

or  R  :  CB  :  :  sin.  C  :  BA. 

Also,  CE  :  CB  :  :  CF  :  CA, 

or  R  :  CB  :  :  cos.  C  :  CA. 

THEOREM  II. 

(42.)  In  any  right-angled  triangle,  radius  is  to  either  side 
as  the  tangent  of  the  adjacent  acute  angle  is  to  the  opposite 
side,  or  the  secant  of  the  same  angle  to  the  hypothenuse. 

Let  the  triangle  CAB  be  right  angled 
at  A,  then  will 

R  :  CA  : :  tang.  C  :  AB  : :  sec.  C  :  CB. 

From  the  point  C  as  a  center,  with  a 

radius  equal  to  the  radius  of  the  tables,    c 


PLANE   TRIGONOMETRY.  tt3 

V 

describe  the  arc  DE,  and  from  the  point  D  draw  DF  perpen- 
dicular to  CA.     Then  DF  will  be  the  tangent,  and  CF  the  se- 
cant of  the  angle  C.     Because  the  triangles  CAB,  CDF  ar<« 
similar,  we  have    CD  :  CA  :  :  DF  :  AB, 
or  R  :  CA  :  :  tang.  C  :  AB. 

Also,  CD  :  CA  :  :  CF  :  CB, 

or  R  :  CA  : :  sec.  C  :  CB. 

(43.)  In  every  plane  triangle  there  are  six  parts  :  three  sides 
and  three  angles.  Of  these,  any  three  being  given,  provided 
one  of  them  is  a  side,  the  others  may  be  determined.  In  a 
right-angled  triangle,  one  of  the  six  parts,  viz.,  the  right  angle, 
is  always  given ;  and  if  one  of  the  acute  angles  is  given,  the 
other  is,  of  course,  known.  Hence  the  number  of  parts  to  be 
considered  in  a  right-angled  triangle  is  reduced  to  four,  any 
two  of  which  being  given,  the  others  may  be  found. 

It.  is  desirable  to  have  appropriate  names  by  which  to  des- 
ignate each  of  the  parts  of  a  triangle.  One  of  the  sides  ad- 
jacent to  the  right  angle  being  called  the  base,  the  other  side 
adjacent  to  the  right  angle  may  be  called  the  perpendicular. 
The  three  sides  will  then  be  called  the  hypothenuse,  base,  and 
perpendicular.  The  base  and  perpendicular  are  sometimes 
called  the  legs  of  the  triangle.  Of  the  two  acute  angles,  that 
which  is  adjacent  to  the  base  may  be  called  the  angle  at  the 
base,  and  the  other  the  angle  at  the  perpendicular. 

We  may,  therefore,  have  four  cases,  according  as  there  are 
given, 

1.  The  hypothenuse  and  the  angles  ; 

2.  The  hypothenuse  and  a  leg ; 

3.  One  leg  and  the  angles  ;  or, 

4.  The  two  legs. 

All  of  these  cases  may  be  solved  by  the  two  preceding  theo- 
rems. 

CASE  I. 

(44.)  Given  the  hypothenuse  and  the  angles,  to  find  the  base 
and  perpendicular. 

This  case  is  solved  by  Theorem  I. 

Radius  :  hypothenuse  : :  sine  of  the  angle  at  the  base  :  per- 
pendicular ; 

: :  cosine  of  the  angle  at  the  base  :  base 
C 


34  TRIGONOMETRY. 

Ex.  1.  Given  the  hypothenuse  275,  and  the  angle  at  the  "base 
57°  23',  to  find  the  base  and  perpendicular. 
The  natural  sine  of  57°  23'  is  .842296  ; 
"  cosine        "  .539016. 

Hence  1  :  275  :  :  .842296  :  231.631=AB. 
1  :  275  :  :  .539016  :  148.229=AC. 
The  computation  is  here  made  by  natural 
numbers.     If  we  work  the  proportion  by  loga-    ( 
rithms,  we  shall  have 

Radius,  10.000000 

Is  to  the  hypothenuse  275  2.439333 

As  the  sine  of  C  57°  23'  9.925465 

To  the  perpendicular  231.63  2.364798. 

Also,  Radius,  10.000000 

Is  to  the  hypothenuse  275  2.439333 

As  the  cosine  of  C  57°  23'  9.731602 

To  the  base  148.23  2.170935, 

Ex.  2.  Given  the  hypothenuse  67.43,  and  the  angle  at  th« 
perpendicular  38°  43',  to  find  the  base  and  perpendicular. 

Ans.  The  base  is  42.175,  and  perpendicular  52.612. 

The  student  should  work  this  and  the  following  examples 

both  by  natural  numbers  and  by  logarithms,  until  he  has  made 

himself  perfectly  familiar  with  both  methods.     He  may  then 

employ  either  method,  as  may  appear  to  him  most  expeditious 

CASE  II. 

(45.)  Given  the  hypothenuse  and  one  leg,  to  find  the  angles 
and  the  other  leg. 

This  case  is  solved  by  Theorem  I. 

Hypothenuse  :  radius  : :  base  :  cosine  of  the  angle  at  the  base. 
Radius  :  hypothenuse  : :  sine  of  the  angle  at  the  base  : 
perpendicular. 

When  the  perpendicular  is  given,  perpendicular  must  be 
substituted  for  base  in  this  proportion. 

Ex.  1.  Given  the  hypothenuse  54.32,  and  the  base  32.11,  tt- 
find  the  angles  and  the  perpendicular. 

Bv  natural  numbers,  we  have 


PLANF    TRIGONOMETRY.  35 

54.32  :  1  :  :  32.11  :  .591127,  which  is  the  cosine  of  53°  45 
47",  the  angl  •.  at  the  base. 

Also,  1  :  54.32  :  :  .806580  :  43.813=the  perpendicular. 

The  computation  may  be  performed  more  expeditiously  b>f 
logarithms,  as  in  the  former  case. 

Ex.  2.  Given  the  hypothenuse  332.49,  and  the  perpendicu- 
lar 98.399,  to  find  the  angles  and  the  base. 

Ans.  The  angles  are  17°  12'  51"  and  72°  47'  9" ;  the  base, 
317.6. 

CASE  III. 

(46.)  Given  one  leg  and  the  angles,  to  find  the  other  leg 
and  hypothenuse. 

This  case  is  solved  by  Theorem  II. 

Radius  :  base  : :  tangent  of  the  angle  at  the  base  :  the  perpen 
dicular. 

: :  secant  of  the  angle  at  the  base  :  hypothenuse. 

When  the  perpendicular  is  given,  perpendicular  must  be 
substituted  for  base  in  this  proportion. 

Ex.  1.  Griven  the  base  222,  and  the  angle  at  the  base  25°  15', 
to  find  the  perpendicular  and  hypothenuse. 

By  natural  numbers,  we  have 

1  :  222  :  :    .471631  :  104.70,  perpendicular ; 
:  :  1.105638  :  245.45,  hypothenuse. 

The  computation  should  also  be  performed  by  logarithms,  SLS 
in  Case  I. 

Ex.  2.  (riven  the  perpendicular  125,  and  the  angle  at  the 
perpendicular  51°  19',  to  find  the  hypothenuse  and  base. 

Ans.  Hypothenuse,  199.99 ;  base,  156.12. 

CASE  IY. 

(47.)  Given  the  two  legs,  to  find  the  angles  and  hypothenuse. 

This  case  is  solved  by  Theorem  II. 

Base  :  radius  :  -.perpendicular :  tangent  of  the  angle  at  the  base. 
Radius  :  base : :  secant  of  the  angle  at  the  base  :  hypothenuse. 

Ex.  1.  Griven  the  base  123,  and  perpendicular  765,  to  find 
the  angles  and  hypothenuse. 

By  natural  numbers,  we  have 

123  :  1  : :  765  :  6.219512,  which  is  the  tangent  of  80°  51 
•*)7",  the  angle  at  the  base. 


36  TRIGONOMETRY. 

1  :  123  :  :  6.299338  :  774.82,  hypothenuse. 

The  computation  may  also  be  made  by  logarithms,  as  iz 
Case  I. 

Ex.  2.  Given  the  base  53,  and  perpendicular  67,  to  find  the 
angles  and  hypothenuse. 

Ans.  The  angles  are  51°  39'  16"  and  38°  20'  44"  ;  hypothe- 
nuse,  85.428. 

Examples  for  Practice. 

1.  Given  the  base  777,  and  perpendicular  345,  to  find  the 
hypothenuse  and  angles. 

This  example,  it  will  be  seen,  falls  under  Case  IV. 

2.  Given  the  hypothenuse  324,  and  the  angle  at  the  base 
48°  17',  to  find  the  base  and  perpendicular. 

3.  Given  the  perpendicular  543,  and  the  angle  at  the  base 
72°  45',  to  find  the  hypothenuse  and  base. 

4.  Given  the  hypothenuse  666,  and  base  432,  to  find  the  an- 
gles and  perpendicular. 

5.  Given  the  base  634,  and  the  angle  at  the  base  53°  27',  to 
find  the  hypothenuse  and  perpendicular. 

6.  Given  the  hypothenuse  1234,  and  perpendicular  555,  to 
find  the  base  and  angles. 

(48.)  "When  two  sides  of  a  right-angled  triangle  are  given, 
the  third  may  be  found  by  means  of  the  property  that  the 
square  of  the  hypothenuse  is  equal  to  the  sum  of  the  squares 
of  the  other  two  sides. 

Hence,  representing  the  hypothenuse,  base,  and  perpendicu 
lar  by  the  initial  letters  of  these  words,  we  have 

A=-/FHhp*;  &=VT^7;  p=  /F=F. 

Ex.  1.  If  the  base  is  2720,  and  the  perpendicular  3104,  what 
is  the  hypothenuse  ?  Ans.,  4127.1. 

Ex.  2.  If  the  hypothenuse  is  514,  and  the  perpendicular  432, 
what  is  the  base  ? 

SOLUTIONS  OF  OBLIQUE-ANGLED  TRIANGLES. 

THEOREM  I. 

(49.)  In  any  plane,  triangle,  the  sines  of  the  angles  are, 
proportional  to  the  opposite  sides. 


PLANE    TRIGONOMETRY.  37 

Let  ABC  be  any  triangle,  and  from  one 
of  its  angles,  as  C,  let  CD  be  drawn  per- 
pendicular to  AB.  Then,  because  the 
triangle  ACD  is  right  angled  at  D,  we 
have  AD  u 

R  :  sin.  A  : :  AC  :  CD;  whence  RxCD^sin.  AxAC. 

For  the  same  reason, 
R  :  sin.  B  :  :  BC  :  CD ;  whence  RxCD=sin.  BxBC. 

Therefore,          sin.  A X  AC = sin.  BxBC, 
or  sin.  A  :  sin.  B  :  :  BC  :  AC. 

THEOREM  II. 

(50.)  In  any  plane  triangle,  the  sum  of  any  two  sides  is  to 
their  difference,  as  the  tangent  of  half  the  sum  of  the  opposite 
angles  is  to  the  tangent  of  half  their  difference. 

Let  ABC  be  any  triangle  ;  then  will 

CB+CA  :  CB-CA  :  :  tang.  ^-^- :  tang.  -^-. 

£  £ 

Produce  AC  to  D,  making  CD  equal  to  CB,  and  join  Di3. 
Take  CE  equal  to  CA,  draw  AE,  and  produce  it  to  F.  Then 
AD  is  the  sum  of  CB  and  CA,  and  BE  is  their  difference. 

The  sum  of  the  two  angles  CAE,  CEA,  is  equal  to  the  sum 
of  CAB,  CBA,  each  being  the  supplement  of  ACB  (Geom., 
Prop.  27,  B.  I.).  But,  since  CA  is  equal 
to  CE,  the  angle  CAE  is  equal  to  the  an- 
gle CEA;  therefore,  CAE  is  the  half 
sum  of  the  angles  CAB,  CBA.  Also,  if 
from  the  greater  of  the  two  angles  CAB, 
CBA,  there  be  taken  their  half  sum,  the 
remainder,  FAB,  will  be  their  half  differ- 
ence (Algebra,  p.  68). 

Since  CD  is  equal  to  CB,  the  angle  ADF 
is  equal  to  the  angle  EBF ;  also,  the  an- 
gle CAE  is  equal  to  ABC,  which  is  equal 
to  the  vertical  angle  BEF.  Therefore,  the  two  triangles  DAF, 
BEF,  are  mutually  equiangular ;  hence  the  two  angles  at  F 
are  equal,  and  AF  is  perpendicular  to  DB.  If,  then,  AF  be 
made  radius,  DF  will  be  the  tangent  of  DAF,  and  BF  will  be 
the  tangent  of  BAF.  But,  by  similar  triangles,  we  have 


88  T  R  I  0  0  N  0  M  E  T  R  Y. 

AD  :  BE  :  :  DF  :  BF ;  that  is, 
CB-I-CA  :  CB-CA  :  :  tang.  — £?  :  tang. 


THEOREM  III. 

(51.)  If  from  any  angle  of  a  triangle  a  perpendicular  ot 
drawn  to  the  opposite  side  or  base,  the  whole  base  will  be  to 
the  sum  of  the  other  two  sides,  as  the  difference  of  those  two 
sides  is  to  the  difference  of  the  segments  of  the  base. 

For  demonstration,  see  Geometry,  Prop.  31,  Cor.,  B.  IY. 

(52).  In  every  plane  triangle,  three  parts  must  be  given  to 
enable  us  to  determine  the  others  ;  and  of  the  given  parts,  one, 
at  least,  must  be  a  side.  For  if  the  angles  only  are  given, 
these  might  belong  to  an  infinite  number  of  different  triangles 
In  solving  oblique-angled  triangles,  four  different  cas^s  mav 
therefore  be  presented.  There  may  be  given, 

1.  Two  angles  and  a  side ; 

2.  Two  sides  and  an  angle  opposite  one  of  them ; 

3.  Two  sides  and  the  included  angle  ;  or, 

4.  The  three  sides. 

We  shall  represent  the  three  angles  of  the  proposed  triangle 
by  A,  B,  C,  and  the  sides  opposite  them,  respectively,  by  a,  b,  c 

CASE  I. 

(53.)  Given  two  angles  and  a  side,  to  find  the  third  angle 
and  the  other  two  sides. 

To  find  the  third  angle,  add  the  given  angles  together,  and 
subtract  their  sum  from  180°. 

The  required  sides  may  be  found  by  Theorem  I.  The  pro- 
portion will  be, 

The  sine  of  the  angle  opposite  the  given  side  :  the  given  side 

:  :  the  sine  of  the  angle  opposite  the  required  side  :  the  re* 
quired  side. 

Ex.  1.  In  the  triangle  ABC,  there  are 
given  the  angle  A,  57°  15',  the  angle  B, 
35°  30',  and  the  side  c,  364,  to  find  the 
other  parts. 

The  sum  of  the  given  angles,  subtracted  A 


PLANE    TRIGONOMETRY.  ^9 

from  180°,  leaves  87°  15'  for  the  angle  C.     Then,  to  find  the 
dde  a,  we  say,         sin.  C  :  c  :  :  sin.  A  :  a. 
By  natural  numbers, 

.998848  :  364  : :  .841039  :  306.49= a. 
This  proportion  is  most  easily  worked  "by  logarithms,  thus , 
As  the  sine  of  the  angle  C,  87°  15',  cornp.,  0.000500 
Is  to  the  side  c,  364,  2.561101 

So  is  the  sine  of  the  angle  A,  57°  15',       9.924816 
To  the  side  a,  306.49,  2.486417. 

To  find  the  side  b  : 

sin.  C  ;  c  :  :  sin.  B  :  b. 
By  natural  numbers, 

.998848  :  364  : :  .580703  :  211.62=*. 
The  work  by  logarithms  is  as  follows  : 

sin.  C,  87°  15',  comp.,  0.000500 

:  c,  364,  2.561101 

:  :  sin.  B,  35°  30',  9.763954 

:  b,  211.62,  2.325555. 

Ex.  2.  In  the  triangle  ABC,  there  are  given  the  angle  A, 
49°  25',  the  angle  C,  63°  48',  and  the  side  c,  275,  to  find  the 
other  parts.  Ans.,  B=66°  47' ;  a=232.766  ;  £=281.67. 

CASE  II. 

(54.)  Given  two  sides  and  an  angle  opposite  one  of  them, 
to  find  the  third  side  and  the  remaining  angles. 

One  of  the  required  angles  is  found  by  Theorem  I.  The 
proportion  is, 

The  side  opposite  the  given  angle  :  the  sine  of  that  angle 

: :  the  other  given  side  :  the  sine  of  the  opposite  angle. 

The  third  angle  is  found  by  subtracting  the  sum  of  the  other 
two  from  180°  ;  and  the  third  side  is  found  as  in  Case  I. 

If  the  side  BC,  opposite  the  given  an-  r 

gle  A,  is  shorter  than  the  other  given  side 
AC,  the  solution  will  be  ambiguous  ;  that 
is,  two  different  triangles,  ABC,  AB'C, 

may  be  formed,  each  of  which  will  satisfy  A      B"'*" "'B' 

the  conditions  of  the  problem. 

The  numerical  result  is  also  ambiguous,  for  the  fourth  term 


40  TRIGONOMETRY 

of  the  fir 3t  proportion  is  a  sine  of  an  angle.     But  this  may  "be 
the  sine  either  of  the  acute  angle  AB'C,  or                     Q 
of  its  supplement,  the  obtuse  angle  ABC 
(Art.  27).    In  practice,  however,  there  will 
generally  be  some  circumstance  to  determ- 
ine whether  the  required  angle  is  acute  or  A.      B  ""  B' 

obtuse.  If  the  given  angle  is  obtuse,  there  can  be  no  ambi 
guity  in  the  solution,  for  then  the  remaining  angles  must  of 
course  be  acute. 

Ex.  1.  In  a  triangle,  ABC,  there  are  given  AC,  458,  BC 
307,  and  the  angle  A,  28°  45',  to  find  the  other  parts. 
To  find  the  angle  B  : 

BC  :  sin.  A  :  :  AC  :  sin.  B. 
By  natural  numbers, 

307  :  .480989  :  :  458  :  .717566,  sin.  B,  the  arc  correspond 
•ng  to  which  is  45°  51'  14",  or  134°  8'  46". 

This  proportion  is  most  easily  worked  by  logarithms,  thus  • 
BC,  307,  comp.,  7.512862 

:  sin.  A,  28°  45',  9.682135 

: :  AC,  458,  2.660865 

:  sin.  B,  45°  51'  14",  or  134°  8'  46",  9.8*5862. 
The  angle  ABC  is  134°  8'  46",  and  the  angie  AB'C,  45°  5  . 
14".     Hence  the  angle  ACB  is  17°  6'  14",  and  the  angle  ACB 
1 05°  23' 46". 
To  find  the  side  AB  : 

sin.  A  :  CB  :  :  sin.  ACB  :  AB. 
By  logarithms, 

sin.  A,  28°  45',  comp.,  0  317865 

•     :  CB,  307,  2.487138 

: :  sin.  ACB,  17°  6'  14",        9.468502 

:  AB,  187.72,  2.273505. 

To  find  the  side  AB' : 

sin.  A  :  CB'  : :  sin.  ACB  :  AB'. 
By  logarithms, 

sin.  A,  28°  45',  comp.,  0.317865 

:  CB',  307,  2.487138 

: :  sin.  ACB',  105°  23'  46",    9.984128 
:  AB',  615.36,  2/789131. 


PLANE    TRIGONOMETRY.  41 

Ex.  2.  In  a  triangle,  ABC,  there  are  given  AB,  532,  BC, 
358,  and  the  angle  C,  107°  40',  to  find  the  other  parts. 

Ans.  A=39°  52'  52" ;  B=32°  27'  8" ;  AC=299.6.  * 

In  this  example  there  is  no  ambiguity,  because  the  givei< 
angle  is  obtuse. 

CASE  III. 

(55.)  Given  two  sides  and  the  included  angle,  to  find  the 
third  side  and  the  remaining  angles. 

The  sum  of  the  required  angles  is  found  by  subtracting  the 
given  angle  from  180°.  The  difference  of  the  required  angles 
is  then  found  by  Theorem  II.  Half  the  difference  added  to 
half  the  sum  gives  the  greater  angle,  and,  subtracted,  gives 
the  less  angle.  The  third  side  is  then  found  by  Theorem  I. 

Ex.  1.  In  the  triangle  ABC,  the  angle  A  is  given  53°  8' , 
the  side  c,  420,  and  the  side  b,  535,  to  find  the  remaining  parts. 

The  sum  of  the  angles  B  +  C=180°-53°  8'=126°  52'. 
Half  their  sum  is  63°  26'. 

Then,  by  Theorem  II., 

535+420  :  535-420  : :  tang.  63C  26'  :  tang.  13°  32  25", 
which  is  half  the  difference  of  the  two  required  angles. 

Hence  the  angle  B  is  76°  58'  25",  and  the  angle  C,  49° 
53  35". 

To  find  the  side  a  : 

sin.  C  :  c  :  :  sin.  A  :  a=439.32. 

Ex.  2.  Given  the  side  c,  176,  a,  133,  and  the  included  angle 
B,  73°,  to  find  the  remaining  parts. 

Ans.,  6=187.022,  the  angle  C,  64°  9'  3",  and  A,  42°  50'  57". 

CASE  IV. 

(56.)  Given  the  three  sides,  to  find  the  angles. 

Let  fall  a  perpendicular  upon  the  longest  side  from  the  op- 
posite  angle,  dividing  the  given  triangle  into  two  right-angled 
triangles.  The  two  segments  of  the  base  may  be  found  bv 
Theorem  III.  There  will  then  be  given  the  hypothenuse  and 
one  side  of  a  right-angled  triangle  to  find  the  angl  r,s. 

Ex.  1.  In  the  triangle  ABC,  the  side  a  is  261,  the  side  b< 
345,  and  c,  395.  What  are  the  angles  ? 

Let  fall  the  perpendicular  CT)  npon  AB. 


42  TRIGONOMETRY. 

Then,  by  Theorem  III., 

AB  :  AC  +  CB  : :  AC-CB  :  AD-  DB ; 
or  395  :  606  : :  84  :  128.87. 

Half  the  difference  of  the  segments  added  to  half  their  sum 
gives  the  greater  segment,  and  subtracted  gives  the  less  seg- 
ment. 

Therefore,  AD  is  261.935,   and  BD, 
133.065. 

Then,  in  each  of  the  right-angled  tri- 
angles, ACD,  BCD,  we  have  given  the  A  D  ~~~B 
hypothenuse  and  "base,  to  find  the  angles  by  Case  II.  of  right- 
angled  triangles.  Hence 

AC  :  R  : :  AD  :  cos.  A=40°  36'  13" ; 
BC  :  R,  : :  BD  :  cos.  B=59°  20'  52' . 
Therefore  the  angle  C=80°  2'  55". 

Ex.  2.  If  the  three  sides  of  a  triangle  are  150,  140,  and  130, 
what  are  the  angles  ? 

Ans.,  67°  22'  48",  59°  29'  23",  and  53°  7'  49" 

Examples  for  Practice. 

L  Given  two  sides  of,  a  triangle,  478  and  567,  and  the  in 
eluded  angle,  47°  30',  to  find  the  remaining  parts. 

2.  Given  the  angle  A,  56°  34',  the  opposite  side,  a,  735,  and 
the  side  £,  576,  to  find  the  remaining  parts. 

3.  Given  the  angle  A,  65°  40',  the  angle  B,  74°  20',  and  the 
side  a,  275,  to  find  the  remaining  parts. 

4.  Given  the  three  sides,  742,  657,  and  379,  to  find  the  an- 
gles. 

5.  Given  the  angle  A,  116°  32',  the  opposite  side,  a,  492, 
and  the  side  c,  295,  to  find  the  remaining  parts. 

6.  Given  the  angle  C,  56°  18',  the  opposite  side,  c,  184,  and 
the  side  b,  219,  to  find  the  remaining  parts. 

This  problem  admits  of  two  answers. 

INSTRUMENTS  USED  IN  DRAWING. 

(57.)  The  following  are  some  of  the  most  important  instru- 
ments used  in  drawing. 

I.  The  dividers  consist  of  two  legs,  revolving  upon  a  pivot 
at  o*ie  extremity.  The  joints  should  be  composed  of  two  dif- 


PLANE    1-RlGONOMElRY. 


ferent  metals,  of  unequal  hardness  :  one  part,  for  example,  ol 
steel,  and  the  other  of 
brass  or  silver,  in  order 
that  they  may  move  upon 
each  other  with  greater 
freedom.  The  points  should  be  of  tempered  steel,  and  when 
the  dividers  are  closed,  they  should  meet  with  great  exactness. 
The  dividers  are  often  furnished  with  various  appendages, 
which  are  exceedingly  convenient  in  drawing.  Sometimes  one 
of  the  legs  is  furnished  with  an  adjusting  screw,  by  which  a 
slow  motion  may  be  given  to  one  of  the  points,  in  which  case 
they  are  called  hair  compasses.  It  is  also  useful  to  have  a 
movable  leg,  which  may  be  removed  at  pleasure,  and  other 
parts  fitted  to  its  place ;  as,  for  example,  a  long  beam  for 
drawing  large  circles,  a  pencil  point  for  drawing  circles  with 
a  pencil,  an  ink  point  for  drawing  black  circles,  &c. 

(58.)  II.  The  parallel  rule  consists  of  two  flat  rules,  made 
of  wood  or  ivory,  and  connected  together  by  two  cross-bars  of 


equal  length,  and  parallel  to  each  other.  This  instrument  is 
useful  for  drawing  a  line  parallel  to  a  given  line,  through  a 
given  point.  For  this  purpose,  place  the  edge  of  one  of  the 
flat  rules  against  the  given  line,  and  move  the  other  rule  until 
;ts  edge  coincides  with  the  given  point.  A  line  drawn  along 
its  edge  will  be  parallel  to  the  given  line. 

(59.)  III.  The  protractor  is  used  to  lay  down  or  to  measure 
angles.  It  consists  of  a  sem- 
icircle, usually  of  brass,  and 
is  divided  into  degrees,  and 
sometimes  smaller  portions, 
the  center  of  the  circle  be- 
ing indicated  by  a  small 


notch. 

To  lay  down  an  angle  with  the  protractor,  draw  a  base  line, 
and  apply  to  it  the  edge  of  the  protractor,  so  that  its  center 
shall  fall  at  tho  angular  point  Count  the  degrees  contained 


TRIGONOMETRY. 


in  the  proposed  angle  on  the  limb  of  the  circle,  and  mark  the 
extremity  of  the  arc  with  a  fine  dot.  Remove  the  instrument, 
and  through  the  dot  draw  a  line  from  the  angular  point ;  it 
will  give  the  angle  required.  In  a  similar  manner,  the  in- 
clination of  any  two  lines  may  be  measured  with  the  pro- 
tractor. 

(60.)  IV.  The  plane  scale  is  a  ruler,  frequently  two  feet  in 
length,  containing  a  line  of  equal  parts,  chords,  sines,  tan 
gents,  &c.  For  a  scale  of  equal  parts,  a  line  is  divided  int. 
inches  and  tenths  of  an  inch,  or  half  inches  and  twentieths. 
When  smaller  fractions  are  required,  they  are  obtained  by 
means  of  the  diagonal  scale,  which  is  constructed  in  the  fol- 
lowing manner.  Describe  a  square  inch,  ABCD,  and  divido 
4  3  2  1  A  .2 .4 .6 .8  B 


DE 

each  of  its  sides  into  ten  equal  parts.  Draw  diagonal  lines 
from  the  first  point  of  division  on  the  upper  line,  to  the  second 
on  the  lower ;  from  the  second  on  the  upper  line,  to  the  thirc 
on  the  lower,  and  so  on.  Draw,  also,  other  lines  parallel  t; 
AB,  through  the  points  of  division  of  BC.  Then,  in  the  trian- 
gle ADE,  the  base,  DE,  is  one  tenth  of  an  inch ;  and,  since 
the  line  AD  is  divided  into  ten  equal  parts,  and  through  the 
points  of  division  lines  are  drawn  parallel  to  the  base,  forming 
nine  smaller  triangles,  the  base  of  the  least  is  one  tenth  of  DE, 
that  is,  .01  of  an  inch ;  the  base  of  the  second  is  .02  of  an  inch ; 
the  third,  .03,  and  so  on.  Thus  the  diagonal  scale  furnishes 
us  hundredths  of  an  inch.  To  take  off  from  the  scale  a  line 
of  given  length,  as,  for  example,  4.45  inches,  place  one  foot  of 
the  dividers  at  F,  on  the  sixth  horizontal  line,  and  extend  the, 
other  foot  to  Gr,  the  fifth  diagonal  line. 

A  half  inch  or  less  is  frequently  subdivided  in  the  samo 
manner. 

(61.)  A  line  of  chords,  commonly  marked  CHO.,  is  found  on 
most  plane  scales,  and  is  useful  in  setting  off  angles.  To  form 
this  line,  describe  a  circle  with  any  convenient  radius,  and  di- 
vide the  circumference  into  degrees.  Let  the  length  of  the 


PLANE   TRIGONOMETRY. 


chords  foi  every  degree  of  the  quadrant  be  determined  and  laid 
off  on  a  scale  :  this  is  called  a  line  of  chords. 

Since  the  chord  of  60°  is  equal  to  radius,  in  order  to  lay 


Sines 


,9o        40        :lo       60       70       80     90 


30       4O     50    fio  70  3\0       30 


down  an  angle,  we  take  from  the  scale  the  chord  of  60°,  and 
with  this  radius  describe  an  arc  of  a  circle.  Then  take  from 
the  scale  the  chord  of  the  given  angle,  and  set  it  off  upon  the 
former  arc.  Through  these  two  points  of  division  draw  lines 
to  the  center  of  the  circle,  and  they  will  contain  the  required 
angle. 

The  line  of  sines,  commonly  marked  SIN.,  exhibits  the  lengths, 
of  the  sines  to  every  degree  of  the  quadrant,  to  the  same  ra- 
dius as  the  line  of  chords.  The  line  of  tangents  and  the  line 
of  secants  are  constructed  in  the  same  manner.  Since  the  sine 
of  90°  is  equal  to  radius,  and  the  secant  of  0°  is  the  same,  the 
graduation  on  the  line  of  secants  begins  where  the  line  of  sines 
ends. 

On  the  back  side  of  the  plane  scale  are  often  found  lines  rep 
resenting  the  logarithms  of  numbers,  sines,  tangents,  &c.  This 
is  called  Grunter's  Scale. 

(62.)  Y.  The  Sector  is  a  very  convenient  instrument  in 
drawing.  It  consists  of 
two  equal  arms,  mova- 
ble about  a  pivot  as  a 
center,  having  several 
scales  drawn  on  the 
faces,  some  single,  oth- 
ers double.  The  single  scales  are  like  those  upon  a  common 
Grunter's  scale.  The  double  scales  are  those  which  proceed 
from  the  center,  each  being  laid  twice  on  the  same  face  of  the 
instrument,  viz.,  once  on  each  leg.  The  double  scales  are  a 
gcale  of  lines,  marked  Lin.  or  L. ;  the  scale  of  chords,  sines, 
&c.  On  each  arm  of  the  sector  there  is  a  diagonal  line,  which 
diverges  from  the  central  point  like  the  radius  of  a  circle,  and 
these  diagonal  lines  are  divided  into  equal  parts. 

The  advantage  of  the  sector  is  to  enable  us  to  draw  a  line 


4(5  TRIGONOMETRY 

upon  paper  to  any  scale ;  as,  for  example,  a  scale  of  6  feet  to 
the  inch.  For  this  purpose,  take  an  inch  with  the  dividers 
from  the  scale  of  inches  ;  then,  placing  one  foot  of  the  dividers 
at  6  on  one  arm  of  the  sector,  open  the  sector  until  the  other 
foot  reaches  to  the  same  number  on  the  other  arm.  Now,  re- 
garding the  lines  on  the 
sector  as  the  sides  of  a 
triangle,  of  which  the 
line  measured  from  6  on 
one  arm  to  6  on  the  oth- 
er arm  is  the  base,  it  is 
plain  that  if  any  other  line  be  measured  across  the  angle  of 
the  sector,  the  bases  of  the  triangles  thus  formed  will  be  pro- 
portional to  their  sides.  Therefore,  a  line  of  7  feet  will  be  rep- 
resented by  the  distance  from  7  to  7,  and  so  on  for  other  lines. 

The  sector  also  contains  a  line  of  chords,  arranged  like  the 
line  of  equal  parts  already  mentioned.  Two  lines  of  chords 
are  drawn,  one  on  each  arm  of  the  sector,  diverging  from  the 
central  point.  This  double  line  of  chords  is  more  convenient 
than  the  single  one  upon  the  plane  scale,  because  it  furnishes 
chords  to  any  radius.  If  it  be  required  to  lay  down  any  angle, 
as,  for  example,  an  angle  of  25°,  describe  a  circle  with  any 
convenient  radius.  Open  the  sector  oo  that  the  distance  from 
GO  to  60,  on  the  line  of  chords,  shall  be  equal  to  this  radius. 
Then,  preserving  the  same  opening  of  the  sector,  place  one  foot 
of  the  dividers  upon  the  division  25  on  one  scale,  and  extend 
the  other  foot  to  the  same  number  upon  the  other  scale :  this 
distance  will  be  the  chord  of  25  degrees,  which  must  be  set  off 
upon  the  circle  first  described. 

The  lines  of  sines,  tangents,  &c.,  are  arranged  in  the  same 
manner. 

(63.)  By  means  of  the  instruments  now  enumerated,  all  the 
cases  in  Plane  Trigonometry  may  be  solved  mechanically. 
The  sides  and  angles  which  are  given  are  laid  down  accord- 
ing to  the  preceding  directions,  and  the  required  parts  are  then 
measured  from  the  same  scale.  The  student  will  do  well  to 
exercise  himself  upon  the  following  problems  : 

I.  Given  the  angles  and  one  side  of  a  triangle,  to  find,  by 
construction,  the  other  two  sides. 


PLANE    TRIGONOMETRY.  47 

Draw  an  indefinite  straight  line,  and  from  the.  scale  of  equal 
parts  lay  off  a  portion,  AB,  equal  to  the  .given  side.  From 
each  extremity  lay  off  an  angle  equal  to  one  of  the  adjacent  an- 
gles, by  means  of  a  protractor  or  a  scale  of  chords.  Extend 
the  two  lines  till  they  intersect,  and  measure  their  lengths  upon 
the  same  scale  of  equal  parts  which  was 
used  in  laying  off  the  base. 

Ex.  1.  Given  the  angle  A,  45°  30',  the 
angle  B,  35°  20',  and  the  side  AB,  43° 
rods,  to  construct  the  triangle,  and  find 
the  lengths  of  the  sides  AC  and  BC. 

The  triangle  ABC  may  be  constructed  of  any  dimensions 
whatever ;  all  which  is  essential  is  that  its  angles  be  made 
equal  to  the  given  angles.  We  may  construct  the  triangle 
upon  a  scale  of  100  rods  to  an  inch,  in  which  case  the  side  AB 
will  be  represented  by  4.32  inches ;  or  we  may  construct  it 
upon  a  scale  of  200  rods  to  an  inch ;  that  is,  100  rods  to  a  half 
inch,  which  is  very  conveniently  done  from  a  scale  on  which 
a  half  inch  is  divided  like  that  described  in  Art.  60 ;  or  we 
may  use  any  other  scale  at  pleasure.  It  should,  however,  be 
remembered,  that  the  required  sides  must  be  measured  upon 
the  same  scale  as  the  given  sides. 

Ex.  2.  Griven  the  angle  A,  48°,  the  angle  C,  113°,  and  the 
side  AC,  795,  to  construct  the  triangle. 

II.  Given  two  sides  and  an  angle  opposite  one  of  them,  to 
find  the  other  parts. 

Draw  the  side  which  is  adjacent  to  the  given  angle.  From 
one  end  of  it  lay  off  the  given  angle,  and  extend  a  line  indefin- 
itely for  the  required  side.  From  the  pther  extremity  of  the 
first  side,  with  the  remaining  given  side  for  radius,  describe 
an  arc  cutting  the  indefinite  line.  The  point  of  intersection 
will  determine  the  third  angle  of  the  triangle. 

Ex.  1.  G-iven  the  angle  A,  74°  45',  the  side  AC,  432,  and 
the  side  BC,  475,  to  construct  the  triangle,  and  find  the  other 
parts. 

Ex.  2.  Given  the  angle  A,  105°,  the  side  BC,  498,  and  the 
side  AC,  375,  to  construct  the  triangle. 

III.  Given  two  sides  and  the  included  angle,  to  find  the 
parts. 


48  TRIGONOMETRY. 

Draw  one  of  the  given  sides.  From  one  end  of  it  lay  off  the 
given  angle,  and  draw  the  other  given  side,  making  the  re- 
quired angle  with  the  first  side.  Then  connect  the  extremities 
of  the  two  sides,  and  there  will  be  formed  the  triangle  required 

Ex.  1.  Given  the  angle  A,  37°  25',  the  side  AC,  675,  and 
the  side  AB,  417,  to  construct  the  triangle,  and  find  the  othei 
parts. 

Ex.  2.  Given  the  angle  A,  75°,  the  side  AC;  543,  and  the 
side  AB,  721,  to  construct  the  triangle. 

IV.  Given  the  three  sides,  to  find  the  angles. 

Draw  one  of  the  sides  as  a  base ;  and  from  one  extremity 
of  the  base,  with  a  radius  equal  to  the  second  side,  describe 
an  arc  of  a  circle.  From  the  other  end  of  the  base,  with  a 
radius  equal  to  the  third  side,  describe  a  second  arc  intersect- 
ing the  former ;  the  point  of  intersection  will  be  the  third  an- 
gle of  the  triangle. 

Ex.  1.  Given  AB,  678,  AC,  598,  and  BC,  435,  to  find  the 
angles. 

Ex.  2.  Given  the  three  sides  476,  287,  and  354,  to  find  the 
angles. 

Values  of  the  Sines,  Cosines,  fyc.,  of  certain  Angles 
(64.)  "We  propose  now  to  examine  the  changes  which  ths 
Lines,  cosines,  &c.,  undergo  in  the  dif-  D 

fcrent  quadrants  of  a  circle.  Draw 
two  diameters,  AB,  DE,  perpendicu- 
lar to  each  other,  and  suppose  one  of 
them  to  occupy  a  horizontal  position, 
the  other  a  vertical.  ,  The  angle  ACD 
is  called  the  first  quadrant,  the  angle 
DCB  the  second  quadrant,  the  angle 
BCE  the  third  quadrant,  and  the  an- 
gle EGA  the  fourth  quadrant;  that  is,  the  first  quadrant  is 
above  the  horizontal  diameter,  and  on  the  right  of  the  vertical 
diameter ;  the  second  quadrant  is  above  the  horizontal  diame- 
ter, and  on  the  left  of  the  vertical,  and  so  on. 

Suppose  one  extremity  of  the  arc  remains  fixed  in  A,  while 
the  other  extremity,  marked  F,  runs  round  the  entire  circum* 
ference  in  the  direction  ADBE. 


P  L  A  N  E     T  R  I  G  O  N  O  M  E  T  R  Y.  49 

When  the  point  F  is  at  A,  or  when  the  arc  AF  is  zero,  the 
sine  is  zero.  As  the  point  F  advances  toward  D,  the  sine  in- 
creases ;  and  when  the  arc  AF  becomes  45°,  the  triangle  CFG 
being  isosceles,  we  have  FGr  :  CF  :  :  1  :  x/2  (Geom.,  Prop.  11, 
Cor.  3,  B.  IV.) ;  or  sin.  45°  :  R  :  :  1  :  x/2. 

T> 

Hence,  sin.  45°=— =JRv/2. 

\/& 

The  sine  of  30°  is  equal  to  half  radius  (Art.  22).  Also,  since 
sin.  A=  VR8— cos.  2A,  the  sine  of  60°,  which  is  equal  to  the  co- 
sine of  30%  -  VR3-iR2=  Vp^p,  V~3. 

The  arc  AF  continuing  to  increase,  the  sine  also  increases 
till  F  arrives  at  D,  at  which  point  the  sine  is  equal  to  the  ra- 
dius ;  that  is,  the  sine  of  90° =R. 

As  the  point  F  advances  from  D  toward  B,  the  sines  dimin- 
ish, and  become  zero  at  B  ;  that  is,  the  sine  of  180° =0. 

In  the  third  quadrant,  the  sine  increases  again,  becomes 
oqual  to  radius  at  E,  and  is  reduced  to  zero  at  A. 

(65.)  When  the  point  F  is  at  A,  the  cosine  is  equal  to  ra- 
dius. As  the  point  F  advances  toward  D,  the  cosine  decreases, 
and  the  cosine  of  45°= sine  45°=JRV/I2.  The  arc  continuing 
to  increase,  the  cosine  diminishes  till  F  arrives  at  D,  at  which 
point  the  cosine  becomes  equal  to  zero.  The  cosine  in  the  sec- 
ond quadrant  increases,  and  becomes  equal  to  radius  at  B ;  in 
the  third  quadrant  it  decreases,  and  becomes  zero  at  E  ;  in  the 
fourth  quadrant  it  increases  again,  and  becomes  equal  to  ra- 
dius at  A. 

(66.)  The  tangent  begins  with  zero  at  A,  increases  with  the 
arc,  and  at  45°  becomes  equal  to  radius.  As  the  point  F  ap- 
proaches D,  the  tangent  increases  very  rapidly  ;  and  when  the 
difference  between  the  arc  and  90°  is  less  than  any  assignable 
quantity,  the  tangent  is  greater  than  any  assignable  quantity. 
Hence  the  tangent  of  90°  is  said  to  be  infinite. 

In  the  second  quadrant  the  tangent  is  at  first  infinitely  great, 
and  rapidly  diminishes  till  at  B  it  is  reduced  to  zero.  In  the 
third  quadrant  it  increases  again,  becomes  infinite  at  E,  and  is 
reduced  to  zero  at  A. 

The  cotangent  is  equal  to  zero  at  D  and  E,  and  is  infinite  at 
A  and  B. 

(67.)  The  secant,  begins  with  radius  at  A,  increases  through 

I) 


50 


TRIGONOMETRY. 


the  first  quadrant,  and  becomes  infinite  at  D  ;  diminishes  in 

the  second  quadrant,  till  at  B  it  is 

equal  to  the  radius ;  increases  again 

in  the  third  quadrant,  and  becomes 

infinite  at  E  ;  decreases  in  the  fourth 

quadrant,  and  becomes  equal  to  the   B 

radius  at  A. 

The  cosecant  is  equal  to  radius  at 
D  and  E,  and  is  infinite  at  A  and  B. 

(68.)  Let  us  now  consider  the  al- 
gebraic signs  by  which  these  lines  are  to  be  distinguished.  In 
the  first  and  second  quadrants,  the  sines  fall  above  the  diame- 
ter AB,  while  in  the  third  and  fourth  quadrants  they  fall  be- 
low. This  opposition  of  directions  ought  to  be  distinguished 
by  the  algebraic  signs  ;  and  if  one  of  these  directions  is  re- 
garded as  positive,  the  other  ought  to  be  considered  as  nega- 
tive. It  is  generally  agreed  to  consider  those  sines  which  fall 
above  the  horizontal  diameter  as  positive  ;  consequently,  those 
which  fall  below  must  be  regarded  as  negative.  That  is,  the 
sines  are  positive  in  the  first  and  seeond  quadrants,  and  nega- 
tive in  the  third  and  fourth. 

In  the  first  quadrant  the  cosine  falls  on  the  right  of  DE, 
but  in  the  second  quadrant  it  falls  on  the  left.  These  two  lines 
should  obviously  have  opposite  signs,  and  it  is  generally  agreed 
to  consider  those  which  fall  to  the  right  of  the  vertical  diam- 
eter as  positive  ;  consequently,  those  which  fall  to  the  left  must 
be  considered  negative.  That  is,  the  cosines  are  positive  in 
the  first  and  fourth  quadrants,  and  negative  in  the  second  and 
third. 

(69.)  The  signs  of  the  tangents  are  derived  from  those  of 

the  sines  and  cosines.     For  tang.  =— *  (Art.  28).     Hence, 

cos.    v 

when  the  sine  and  cosine  have  like  algebraic  signs,  the  tan- 
gent will  be  positive ;  when  unlike,  negative.    That  is,  the  tan 
gent  is  positive  in  the  first  and  third  quadrants,  and  negative 
in  the  second  and  fourth. 

T>3 

Also,  cotangent  =—    -  (Art.  28) ;  hence  the  tangent  and 
tang. 

>>»tanwent  have  always  the  same  sign. 


PLANE    TRIGONOMETRY.  51 

•pa 

"We  have  seen  that  sec.  = :  hence  the  secant  must  have 

cos. 

the  same  sign  as  the  cosine. 

T>2 

Also,  cosec.  =  - —  ;  hence  the  cosecant  must  have  the  same 
sin.  ' 

sign  as  the  sine. 

(70.)  The  preceding  results  are  exhibited  in  the  following 
tables,  which  should  be  made  perfectly  familiar : 

First  quad.    Second  quad.     Third  quad.    Fourth  quad 

Sine  and  cosecant,  +  + 

Cosine  and  secant,  -t-  -f 

Tangent  and  cotangent,     +  4- 


Sine, 

Cosine, 

Tangent, 

Cotangent, 

Secant, 

Cosecant, 

(71.)  In  Astronomy  we  frequently  have  occasion  to  considei 
arcs  greater  than  360°.  But  if  an  entire  circumference,  or  any 
number  of  circumferences,  be  added  to  any  arc,  it  will  termin- 
ate in  the  same  point  as  before.  Hence,  if  C  represent  an  en- 
tire circumference,  or  360°,  and  A  any  a$G  whatever,  we  shall 
have 

sin.  A=sin.  (C+A)=sin.  (2C+A)=sin.  (3C+A)=,  &c. 

The  same  is  true  of  the  cosine,  tangent,  &c. 

We  generally  consider  those  arcs  as  positive  which  are  esti- 
mated from  A  in  the  direction  ADBE.  If,  then,  an  arc  were 
estimated  in  the  direction  AEBD,  it  should  be  considered  as 
negative ;  that  is,  if  the  arc  AF  be  considered  positive,  AH 
must  be  considered  negative.  But  the  latter  belongs  to  tho 
fourth  quadrant ;  hence  its  sine  is  negative.  Therefore,  sin 
(-A)  =  -sin.A. 

The  cosine  CG  is  the  same  for  both  the  arcs  AF  and  AH. 
Hence,  cos.  (— A)=cos.  A. 

Also,  tang.  ( — A)  —  —tang.  A. 

And  cot.  ( — A)  = — cot.  A. 


0° 

90° 

180° 

270° 

3603 

0 

+R 

0 

-R 

0 

+R 

0 

-R 

0 

+R 

0 

GO 

0 

00 

0 

it,      00 

0 

00 

0 

00 

+R 

00 

-R 

00 

+R 

,       00 

4-R 

GO 

-R 

00 

TRIGONOMETRY. 


TRIGONOMETRICAL  FORMULA. 

(72.)  Expressions  for  the  sine  and  cosine  of  the  sum  and 
difference  of  two  arcs. 

Let  AB  and  BD  represent  any  two  given  arcs ;  take  BE 
equal  to  BD  :  it  is  required  to  find  an         D 
expression  for  the  sine  of  AD,  the  sum, 
and  of  AE,  the  difference  of  these  arcs. 

Put  AB=a,  and  BD=£;  then  AD= 
a+b,  and  AE  —  a— b.    Draw  the  chord 
DE,  and  the  radius  CB,  which  may  be 
represented  by  R.     Since  DB  is  by 
construction  equal  to  BE,  DF  is  equal 
to  FE,  and  therefore  DE  is  perpendic-  ( 
ular  to  CB.     Let  fall  the  perpendiculars  EGr,  BH,  FI,  and  DK 
upon  AC,  and  draw  EL,  FM  parallel  to  AC. 

Because  the  triangles  BCH,  FCI  are  rjirailar,  we  have 
CB  :  CF  :  :  BH  :  FI ;  or  R  :  cos.  b  :  :  sin.  a  :  FI. 

__    sin.  a  cos.  b 

Therefore,  FI= ^ . 

it 

Also,  CB  :  CF  :  :  CH  :  CI ;  or  R  :  cos.  b  :  :  cos.  a  :  CI. 

rr(1       ,  nT    cos.  a  cos,  b 

Therefore,  CI= —  — . 

it 

The  triangles  DFM,  CBH,  having  their  sides  perpendiculai 
each  to  each,  are  similar,  and  give  the  proportions  - 

CB  :  DF  : :  CH  :'  DM  ;  or  R  :  sin.  b  : :  cos.  a  :  DM. 

._    cos.  a  sin.  b 
Hence 


Also,  CB  :  DF  :  :  BH  :  FM ;  or  R  :  sin.  b  :  :  sin.  a  :  FM. 

„.  sin.  a  sin.  b 

Hence  FM= ^ . 

But  FI+DM=DK  =sin.  (a+b) ; 

and  CI-FM-CK  =cos.  (a+b). 

Also,  FI-FL  =  EGr=sin.  (a-b) ; 

and  CI+EL=CG-=cos.  (a-b). 

„  sin.  a  cos.  b+cos.  a  sin.  b  , 

Hence,  sm.  (a+b)=—             — ^ —               -  (1) 


cos.  (a+b)~- 


-LA/         • 

cos.  a  cos  b— sin.  a  sin.  b 


R" 


PLANE    TRIGONOMETRY.  53 

sin.  a  cos.  b — cos.  a  sin.  b 
sin.  («-£)=-  -w-  -   (3) 

cos.  a  cos.  £+sin.  a  sin.  b    . 
cos.  (a-b)=  -^—  —   (4) 

(73.)  Expressions  for  the  sine  and  cosine  of  a  double  arc. 
If,  in  the  formulas  of  the  preceding  article,  we  make  b=  a 
the  first  and  second  will  become 

2  sin.  a  cos.  a 
sin.  2a=—     - , 


cos.  2a= — 


R 

Making  radius  equal  to  unity,  and  substituting  the  values  ol 
sin.  a,  cos.  a.  &c.,  from  Art.  28,  we  obtain 

2  tang,  a 


sin.  2a— 
cos.  2a= 


1+tang.  V 
1— tang.  2& 


1+tang.  V 

(74.)  Expressions  for  the  sine  and  cosine  of  half  a  given 
arc. 

Tf  we  put  \a  for  a  in  the  preceding  equations,  we  obtain 

2  sin.  \a  cos.  \a 
sm.  a=  R         —  , 

cos.  2^#—  sin.  2i& 


_ 

We  may  also  find  the  sine  and  cosine  of  \a  in  terms  of  a. 
Since  the  sum  of  the  squares  of  the  sine  and  cosine  is  equal 
to  the  square  of  radius,  we  have 

cos.  2J#+sin.  2^a=R2. 
And,  from  the  preceding  equation, 

cos.  2|-a—  sin.  2^&=R  cos.  a. 
If  we  subtract  one  of  these  from  the  other,  we  have 

2  sin.  2J#=R2—  R  cos.  a. 
And,  adding  the  same  equations, 

2  cos.  24#=R2+R  cos.  a. 
Hence,  sin.  \a=  v^R2—  ^R  cos.  a; 


cos.  \a—  VJR3+^R  cos.  a. 

(75.)  Expressions  for  the  products  of  sines  and  cosines. 
By  adding  and  subtracting  the  formulas  of  Art.  72,  we  obtain 


54  TRIGONOMETRY. 

2 
sin.  (a  +b)-\-  sin.  (a—  0)=^  sin.  a  cos.  b. 

2 
sin.  (a+b)—  sin.  (&—£)=—  cos.  a  sin.  b; 

Jtw 
O 

cos.  (a+#)+cos.  (a—  ^)—  w-  cos.  a  cos.  b  ; 

2 

cos.  (a—  b)—  cos.  (#+Z>)~:p-  sin.  a  sin.  #. 

1\/ 

If,  in  these  formulas,  we  make  a+b=A,  and  a—  &=B  ;  that 
is,  a—  ^(A+B),  and  &=^(A—  B),  we  shall  have 

2 

sin.  A+sin.  B=^  sin.  i(A+B)  cos.  i(A-B)     (1) 

2 
sin.  A-sin.  B=^-  sin.  J(A-B)  cos.J(A+B)     (2) 

2 
cos.  A+cos.  B=£  cos.  |(A+B)  cos.  i(A-B)     (3) 

cos.  B-cos.  A=|-  sin.  J(A+B)  sin.  J(A-B)     (4) 

(76.)   Dividing  formula  (1)  by  (2),  and  considering  that 

sin.  a    tang,  a 

-  =  —  =2  —  (Art.  28),  we  have 
cos.  a        R       v 

sin.  A+sin.  B_sin.  j(A+B)  cos.  ^(A-B)_tang.  i(A+B)a 
sin.  A-sin.  B~sin.  i(A-B)  cos.  i(A+B)~~tang.  J(A-B)' 

that  is, 

T/ie  swm  o/  ^/ie  sme5  o/  two  arcs  is  to  their  difference,  as 

the  tangent  of  half  the  sum  of  those  arcs  is  to  the  tangent  of 

half  their  difference. 


.Dividing  formula  (3)  by  (4),  and  considering  that  -—  =-^- 

sin       xv 

T> 

T-  —  (Art.  28),  we  have 
tang.  v 

cos.  A+cos.  B_cos.  ^(A+B)  cos.  J(A—B)_  cot. 


cos.  B-cos.  Asin.  i(A+B)  sin.  i(A-Bj~~tang.  J(A-B) 
that  is, 

The  sum  of  the  cosines  of  two  arcs  is  to  their  difference,  as 
the  cotangent  of  half  the  sum  of  those  arcs  is  to  the  tang-*i1 
of  half  their  difference. 


PLANE    TRIGONOMETRY.  55 

From  the  first  formula  of  Art.  74,  by  substituting  A+B  for 
j,  we  have 

sin.  (A+B)=2siM(A+B)Xcoa.i(A+B) 

tt 

Dividing  formula  (1),  Art.  75,  by  this,  we  obtain 
sin.  A+sin.  B_sin.  j(A+B)  cos.  |(A-B)_cos.  ^(A-B)^ 
sin.  (A+BT~sin.  |(A+B)  cos.  i(A+B)~cos.£(A+B)  ; 
that  is, 

The  sum  of  the  sines  of  two  arcs  is  to  the  sine  of  their 
sum,  as  the  cosine  of  half  the  difference  of  those  arcs  is  to 
the  cosine  of  half  their  sum. 

If  we  divide  equation  (1),  Art.  72,  by  equation  (3),  we  sha.  1 
have 

sin.  (#+&)_  sin.  a  cos.  b+cos.  a  sin.  b 
sin.  (a—b)     sin.  a  cos.  b—  cos.  a  sin.  b' 
By  dividing  both  numerator  and  denominator  of  the  second 

tang.        sin. 
member  by  cos.  a  cos.  b,  and  substituting       °   for  —  -,  we  ob- 

sin.  (a+b)    tang,  a+tang.  b 
tain  —  :  ;  that  is, 

sin.  (a—  b)    tang,  a—  tang,  b 

The  sine  of  the  sum  of  two  arcs  is  to  the  sine  of  their  dif- 
ference, as  the  sum  of  the  tangents  of  those  arcs  is  to  the 
difference  of  the  tangents. 

From  equation  (3),  Art.  72,  by  dividing  each  member  by  cos 
a  cos.  b,  we  obtain 

sin.  (a—b)  _sin.  a  cos.  b—  cos.  a  sin.  &_tang.  a—  tang,  b 

cos.  a  cos.  b  R  cos.  a  cos.  b  Ra 

that  is, 

The  sine  of  the  difference  of  two  arcs  is  to  the  product  of 
their  cosines,  as  the  difference  of  their  tangents  is  to  the 
square  of  radius. 

(77.)  Expressions  for  the  tangents  of  arcs. 

If  we  take  the  expression  tang.  (a+b)=—    —j^  —  p~-  (Art. 

28),  and  substitute  for  sin.  (a+b)  and  cos.  (a+b)  their  values 
given  in  Art.  72,  we  shall  find 

R  (sin.  a  cos.  b+cos.  a  sin.  b) 

7 


tang.  :  —  r 

cos.  a  cos.  b—  sin.  a  sm.  b 


TRIGONOMETRY 


cos.  a  tang,  a  cos.  b  tang,  b  ,  t 

But  sin  a=  -  R    6     ,  and  sin.  6=  --  g-     -(Art.  28) 

If  we  substitute  these  values  in  the  preceding  equation,  ami 
divide  all  the  terms  by  cos.  a  cos.  b,  we  shall  have 

tang.  (a+ft)= 


f  . 

—  tang,  a  tang.  6 

In  like  manner  we  shall  find 

7.     R2  (tang,  a—  tang,  b) 

-^R'+tang.tttang.T- 
Suppose  &=<3,  then 

2R2  tang.  « 
tang.  20=:^—      ^-r. 
R  a—  tang.2a 

Suppose  b=2a,  then 

t         3  _R2  (tang,  a  +  tang.  2a) 
R2—  tang,  a  tang.  2a  ' 
In  the  same  manner  we  find 

..     cot.  a  cot.  b—  R2 
cot.  (a+b)=  --  —j—         —  , 
cot.  b+  cot.  a   ' 


_.     cot.  a  cot. 
cot.  (a—  b)=  --  7-7  -  T 

cot.  b—  cot.  a 

(78,)  "When  the  three  sides  of  a  triangle  are  given,  the  an- 
gles may  be  found  by  the  formula 


sm.      = 


be 
where  S  represents  half  the  sum  of  the  sides  a,  b,  and  c. 

Demonstration. 
Let  ABC  be  any  triangle ;  then  (Geom.,  Prop.  12,  B.  IV.), 

AB2+AC'-BC2 


BC2=AB2-.  c 


Hence,      AD= 

But  in  the  right-angled  triangle  ACD, 
we  have 

R  :  AC  :  :  cos.  A  :  AD. 
RxAD 


Hence,  cos.  A=: 

ng  the  value 
cos.  A=Rx 


AC 
or,  by  substituting  the  value  of  AD, 

AB2+AC2-BCa 
2ABXAC 


JPLANE    TRIGONOMETRY.  5? 

Let  a,  b}  c  represent  the  sides  opposite  the  angles  A,  Bj  C  ; 

^2,2  _     2 

Mien,  cos.  A=Rx 


By  Art,  74,  we  have  2  sin.  2^A=R2-R  cos.  A. 
Substituting  for  cos.  A  its  value  given  above,  we  obtain 


2  sm.2iA=R2-R2X  —  ^  --  =R8 
2bc 


=R8X 


^ 

2bc 

_WX(a+b-c)(a+c-b) 

2bc 
Put  $=^(a+b+c),  and  we  obtain,  after  reduction, 


In  the  same  manner  we  find 


ac 


blx.  1.  What  are  the  angles  of  a  plane  triangle  whose  sides 
are  432,  543,  and  654? 

HereS=814.5;  8-6=382.5;  S-c=271.5. 

log.  382.5  2.582631 

log.  271.5  2.433770 

log.  b,  432  comp.    7.364516 

log.  c,  543  comp.    7.265200 

2)  19.646117 

sin.  JA,  41°  42'  36J".  9.823058. 

Angle  A=83°  25'  13". 

In  a  similar  manner  we  find  the  angle  B=41°  0'  39",  and 
the  angle  0=55°  34'  8". 

Ex.  2.  What  are  the  angles  of  a  plane  triangle  whose  sides 
are  245,  219,  and  91  ? 

(79.)  On  the  computation  of  a  table  of  sines,  cosines,  SfC. 
In  computing  a  table  of  sines  and  cosines,  we  begin  with 
finding  the  sine  and  cosine  of  one  minute,  and  thence  deduce 
the  sines  and  cosines  of  larger  arcs.  The  sine  of  so  small  an 
angle  as  one  minute  is  nearly  equal  to  the  corresponding  aic. 
The  radius  being  taken  as  unity,  the  semicircumference  is 


5S  TRIGONOMETRY. 

known  to  be  3.14159.  This  being  divided  successively  by  180 
and  60,  gives  .0002908882  for  the  arc  of  one  minute,  which 
may  be  regarded  as  the  sine  of  one  minute. 

The  cosine  of  1'=  VI- sin.2= 0.9999999577. 
The  sines  of  very  small  angles  are  nearly  proportional  to  the 
angles  themselves.  We  might  then  obtain  several  other  sines 
by  direct  proportion.  This  method  will  give  the  sines  correct 
to  five  decimal  places,  as  far  as  two  degrees.  By  the  follow- 
ing method  they  may  be  obtained  with  greater  accuracy  for 
the  entire  quadrant. 

By  Art.  75,  we  have,  by  transposition, 

sin.  (a-\-b}  —  2  sin.  a  cos.  b  —  sin.  (a—b), 
cos.  (a+b)=2  cos.  a  cos.  b— cos.  (a— b). 
ff  we  make  a=b,  26,  36,  &c.,  successively,  we  shall  have 
sin.  2b=2  sin.  b  cos.  b  ; 
sin.  36=2  sin.  2b  cos.  6— sin.  b 
sin.  46=2  sin.  36  cos.  b  —  sin.  26, 

&c.,  &c. 

cos.  26=2  cos.  b  cos.  6—1 ; 
cos.  36=2  cos.  26  cos.  6— cos.  6  ; 
cos.  46=2  cos.  36  cos.  6— cos.  26, 

&c.,  &c. 

Whence,  making  6=1',  we  have 

sin.  2' =2  sin.  1'  cos.  1'  =.000582 

sin.  3'=2  sin.  2'  cos.  I'-sin.  1'=.000873; 
sin.  4'=2  sin.  3'  cos.  I'-sin.  2'=.001164, 

&c.,  &c. 

cos.  2'=2  cos.  V  cos.  V-  1  =0.999999  : 
cos.  3'=2  cos.  2'  cos.  I'-cos.  l'=0.999999  ; 
cos.  4'=2  cos.  3'  cos.  I'-cos.  2'=0.999999, 

&c.,  &c. 

The  tangents,  cotangents,  secants,  and  cosecants  are  easily 
from  the  sines  and  3osines,     Thus, 

sin.  1'  cos.  1' 

tans.  1'  = 7-  cot.  !'= 


cos.  1'  sin.  1  ' 

sec.  1'= -r-  ;          cosec.  1'=-^ —  77  ; 

cos.  1  sin.  1 


BCOK  HI. 

MENSURATION  OF  SURFACES. 

(80.)  THE  area  of  a  figure  is  the  space  contained  within  the 
line  or  lines  by  which  it  is  bounded.  This  area  is  determined 
by  finding  how  many  times  the  figure  contains  some  other  sur- 
face, which  is  assumed  as  the  unit  of  measure.  This  unit  is 
commonly  a  square  ;  such  as  a  square  inch,  a  square  foot,  a 
square  rod,  &c. 

The  superficial  unit  has  generally  the  same  name  as  the 
linear  unit,  which  forms  the  side  of  the  square.     Thus, 
the  side  of  a  square  inch  is  a  linear  inch  ; 
"      "    of  a  square  foot  is  a  linear  foot; 
"      "    of  a  square  yard  is  a  linear  yard,  &c. 

There  are  some  superficial  units  which  have  no  correspond- 
ing linear  units  of  the  same  name,  as,  for  example,  an  acre. 

The  following  table  contains  the  square  measures  in  com. 
mon  use : 

Table  of  Square  Measures. 

ty.  Inches.  Sq.  Feet. 

144  =  1  Sq.  Yards. 

1296-  9  =  1       Sq.Rods. 

39204=          272H          30H          1    &OT-L 
627264=        4356  =        484  =        16=       1     Acret. 
6272640=      43560  =      4840  =       160=     10=     1   M 
4014489600=27878400  =3097600  =102400 =6400= 640 =] 

PROBLEM  I. 
(81.)  To  find  the  area  of  a  parallelogram. 

RULE  I. 

Multiply  the  base  by  the  altitude. 

For  the  demonstration  of  this  rule,  see  Geometry,  Prop.  5 
B.  IV. 


60  TRIGONOMETRY. 

Ex.  1.  What  is  the  area  of  a  parallelogram  whose  base  in 
17.5  rods,  and  the  altitude  13  rods  ? 

Ans.,  227.5  square  rods. 

Ex.  2.  What  is  the  area  of  a  square  whose  side  is  315.7 
feet?  Ans.,  99666.49  square  feet. 

Ex.  3.  What  is  the  area  of  a  rectangular  board  whose  length 
is  15.25  feet,  and  breadth  15  inches  ? 

Ans.,  19.0625  square  feet. 

Ex.  4.  How  many  square  yards  are  there  in  the  four  sides 
of  a  room  which  is  18  feet  long,  15  feet  broad,  and  9  feet  high  ? 

Ans.,  66  square  yards. 

(82.)  If  the  sides  and  angles  of  a  parallelogram  are  given, 
the  perpendicular  height  may  be  found  by         D 
Trigonometry.     For  DE  is  one  side  of  a 
right-angled  triangle,  of  which  AD  is  the 
hypothenuse.     Hence, 

K,  :  AD  :  :  sin.  A  :  DE  ;  A      E 

,  .  ,  .„     ADxsin,  A 

from  which  DE= ^ . 

It 

ABxADXsin.  A 
Therefore,  the  amz=ABxDE=—  — . 

it 

Hence  we  derive 

RULE  II. 

Multiply  together  two  adjacent  sides ,  and  the  sine  of  tht, 
included  angle. 

Ex.  1.  What  is  the  area  of  a  parallelogram  having  an  angle 
of  58°,  and  the  including  sides  36  and  25.5  feet  ? 

Ans.  The  area=36x  25.5  X. 84805  (natural  sine  of  58°)== 
778.508  square  feet. 

The  computation  will  generally  be  most  conveniently  per- 
formed by  logarithms. 

Ex.  2.  What  is  the  area  of  a  rhombus,  each  of  whose  sides 
is  21  feet  3  inches,  and  each  of  the  acute  angles  53°  20'  ? 

Ans.,  362.209  feet. 

Ex.  3.  How  many  acres  are  contained  in  a  parallelogram 
one  of  whose  angles  is  30°,  and  the  including  sides  are  25.35 
and  10.4  chains  ?  Ans.,  13  acres,  29.12  rods 


MENSURATION  or  SURFACES.        61 

PROBLEM  II. 
(83.)  To  find  the  area  of  a  triangle. 

RULE  I. 

Multiply  the  base  by  half  the  altitude. 

For  demonstration,  see  Geometry,  Prop.  6,  B.  IYi 

Ex.  1.  How  many  square  yards  are  contained  in  a  triangle 
whose  base  is  49  feet,  and  altitude  25 J  feet  ? 

Ans.,  68.736. 

Ex.  2.  "What  is  the  area  of  a  triangle  whose  base  is  45  feet, 
and  altitude  27.5  feet  ?  Ans.,  618.75  square  feet. 

(84.)  When  two  sides  and  the  included  angle  are  given,  wo 
may  use 

RULE  II. 

Multiply  half  the  product  of  two  sides  by  the  sine  of  the 
included  angle. 

The  reason  of  this  rule  is  obvious,  from  Art.  82,  since  a  tri- 
angle is  half  of  a  parallelogram,  having  the  same  base  and  av 
titude. 

Ex.  1.  What  is  the  area  of  a  triangle  of  which  two  sides  are 
45  and  32  feet,  and  the  included  angle  46°  30'  ? 

Ans.  The  area=45x  16  X. 725374  (natural  sine  of  46°  30')= 

522.269  feet. 

Ex.  2.  What  is  the  area  of  a  triangle  of  which  two  sides  are 
127  and  96  feet,  and  the  included  angle  67°  15'  ? 

Ans. 
(85.)  When  the  three  sides  are  known,  we  may  use 

RULE  III. 

From  half  the  sum  of  the  three  sides  subtract  each  side  sev- 
erally ;  multiply  together  the  half  sum  and  the  three  remain- 
ders, and  extract  the  square  root  of  the  product. 

Demonstration. 

Let  a,  b,  c  denote  the  sides  of  the  tri- 
angle ABC  ;  then,  by  Geometry,  Prop.  12, 
B.  IV.,  we  have  BC2=AB'+AC2-2ABx 
\I),  or  a*=b*-l-c*-2cX AD;  whence,  A  D 


62  TRIGONOMETRY 


But  CD'=AC'-AD'; 

nrv     i.     ( 

hence      CD'=£3- 


2c 
ABxCD 


But  th 

^  """  ^  ^ 

The  quantity  under  the  radical  sign  being  the  difference  ol 
two  squares,  may  be  resolved  into  the  factors  2bc+(b*+c*— a') 
and  2bc—  (62+c'2— a2) ;  and  these,  in  the  same  manner,  may 
be  resolved  into        (b+c+a)X(b+c— a), 
and 


Hence,  if  we  put  S  equal  to  -  -  >  we  sna^  nave 


the  are^= 

Ex.  1.  "What  is  the  area  of  a  triangle  whose  siuds  are  125, 
173,  and  216  feet  ? 

Here  8=257,  S-&=84, 

S-a=132,  S-c=41. 

Hence  the  area=  V257X  132x84x41  =10809  square  feet. 
Ex.  2.  How  many  acres  are  contained  in  a  triangle  whose 
sides  are  49,  50.25,  and  25.69  chains  ? 

Ans.j  61  acres,  1  rood,  39.68  perches. 
Ex.  3.  What  is  the  area  of  a  triangle  whose  sides  are  234, 
289,  and  345  feet? 

Ans. 

(86.)  In  an  equilateral  triangle,  one  of  whose  sides  is  a,  tho 
expression  for  the  area  becomes 


4 

that  is,  the  area  of  an  equilateral  triangle  is  equal  to  on 
fourth  the  square  of  one  of  its  sides  multiplied  by  the  square 
root  of  3. 

Ex.  What  is  the  area  of  a  triangle  whose  sides  are  each  37 
feet?  Ans..  592.79  feet 


MENSURATION  or  SURFACES.        63 

PROBLEM  III. 
(87.)  To  find  the  area  of  a  trapezoid. 

RULE. 

Multiply  half  the  sum  of  the  parallel  sides  into  their  per 
ptndicular  distance. 

For  demonstration,  see  Geometry,  Prop.  7,  B.  IV. 
Ex.  1.  What  is  the  area  of  a  trapezoid  whose  parallel  sidea 
are  156  and  124,  and  the  perpendicular  distance  between  them 
57  feet? 

Ans.,  7980  feet. 

Ex.  2.  How  many  square  yards  in  a  trapezoid  whose  par- 
allel sides  are  678  and  987  feet,  and  altitude  524  feet  ? 

Ans. 

PROBLEM  IV. 
(88.)  To  find  the  area  of  an  irregular  polygon. 

RULE. 

Draw  diagonals  dividing  the  polygon  into  triangles,  and 
find  the  sum  of  the  areas  of  these  triangles. 

Ex.  1.  What  is  the  area  of  a  quadrilateral,  one  of  whose 
diagonals  is  126  feet,  and  the  two  perpendiculars  let  fall  upon 
it  from  the  opposite  angles  are  74  and  28  feet  ? 

Ans.,  6426  feet 

Ex.  2.  In  the  polygon  ABODE,  there 
are  given  EC=205,  EB=242,  AF=65, 
Ca=114,  and  DH^llO,  to  find  the  area. 

Ans. 

1  (89.)  If  the  diagonals  of  a  quadrilateral 
are  given,  the  area  may  be  found  by  the 
following 

RULE. 

Multiply  half  the  product  of  the  diagonals  by  the  sine  of 
the  angle  at  their  intersection. 

Demonstration. 
The  sines  of  the  four  angles  at  E  are  all  equal  tc  each  other 


64 


TRIGONOMETRY. 


B 


since  the  adjacent  angles  AED,  DEO  are  the  supplements  of 
each  other  (Art.  27).     But,  according  to 
the  Rule,  Art.  84,  the  area  of 
the  triangle  ABE=JAExBEXsine  E  ; 
"         "       AED=iAExDEXsine  E; 
«         «       BEC=JBExECxsine  E; 
"         "       DEC=iDExECXsine  E. 
Therefore, 

the  area  of  ABCD=i(AE+EC)x(BE+ED)Xsine  E 

= JAG xBDx sine  E. 

Ex.  1.  If  the  diagonals  of  a  quadrilateral  are  34  and  56 
rods,  and  if  they  intersect  at  an  angle  of  67°,  what  is  the  area? 

Ans.,  876.32. 

Ex.  2.  If  the  diagonals  of  a  quadrilateral  are  75  and  49, 
and  the  angle  of  intersection  is  42°,  what  is  the  are^  ? 

Ans. 

PROBLEM  Y. 
(90.)  To  find  the  area  of  a  regular  polygon. 

RULE  I. 

Multiply  half  the  perimeter  by  the  perpendicular  let  fall 
from  the  center  on  one  of  the  sides. 
For  demonstration,  see  Geometry,  Prop.  7,  B.  VI. 
Ex.  1.  What  is  the  area  of  a  regular  pentagon  whose  side 
is  25,  and  the  perpendicular  from  the  center  17.205  feet  ? 

Ans.,  1075.31  feet. 

Ex.  2.  What  is  the  area  of  a  regular  octagon  whose  side  ia 
53,  and  the  perpendicular  63.977  ? 

Ans. 

(91.)  When  the  perpendicular  is  not  given,  it  may  be  com- 
puted  from  the  perimeter  and  number  of  sides.  If  we  divide 
360  degrees  by  the  number  of  sides  of  the 
polygon,  the  quotient  will  be  the  angle  ACB 
at  the  center,  subtended  by  one  of  the  sides. 
The  perpendicular  CD  bisects  the  side  AB, 
and  the  angle  ACB.  Then,  in  the  triangle 
ACD,  we  have  (Art.  42), 

R  :  AD  :  cot.  ACD  :  CD  ;  that  is, 


MENS    A  ATI  ON    OF    SURFACES.  6f» 

Radius  is  to  half  of  one  of  the  s\  des  of  the  polygon,  as  the 
cotangent  of  the  opposite  angle  is  to  the  perpendicular  from 
tks  center. 

Ex.  3.  Find  the  area  of  a  regular  hexagon  whose  side  is  32 
inches. 

The  angle  ACD  is  TV  of  360°=30G.     Then 
R  :  16  :  :  cot.  30°  :  27.7128=CD,  the  perpendicular  ; 

and  the  area=27.7128x  16x6  =2660.4288. 
Ex.  4.  Find  the  area  of  a  regular  decagon  whose  side  is  4b 
feat.  Ans.,  16280.946. 

(92.)  In  this  manner  was  computed  the  following  table  of 
tlia  areas  of  regular  polygons,  in  which  the  side  of  each  poly- 
is  supposed  to  be  a  unit. 


TABLE  OF  REGULAR  POLYGONS. 

Names.  Sides.  Areas. 

Triangle,  3  0.4330127. 

Square,  4  1.0000000. 

Pentagon,  5  1.7204774. 

Hexagon,  6  2.5980762. 

Heptagon,  7  3.6339124 

Octagon,  8  4.8284271 

Nonagon,  9  6.1818242. 

Decagon,  10  7.6942088. 

Undecagon,  11  9.3656399. 

Dodecagon,  12  11.1961524. 

By  the  aid  of  this  table  may  be  computed  the  area  of  any 
other  regular  polygon  having  not  more  than  twelve  sides.  For, 
since  the  areas  of  similar  polygons  are  as  the  squares  of  their 
homologous  sides,  we  derive 

RULE  II. 

Multiply  the  square  of  one  of  the  sides  of  the  polygon  by 
the  area  of  a  similar  polygon  whose  side  is  unity. 

Ex.  5.  What  is  the  area  of  a  regular  nonagon  whose  side 
is  63  ?  Ans.,  24535.66. 

Ex.  6.  What  is  the  area  of  a  regular  dodecagon  whose  side 
is  54  feet  ?  Ans.,  32647.98  feet. 

E 


fi*>  TRIGONOMETRY. 

PROBLEM  VI 
(93.)  To  find  the  circumference  of  a  circle  from  its  diameter 

RULE 

Multiply  the  diameter  by  3.14159. 

For  the  demonstration  of  this  rule,  see  Geometry,  Prop.  13, 
Cor.  2,  B.VL 

When  the  diameter  of  the  circle  is  small,  and  no  great  ac- 
curacy is  required,  it  may  be  sufficient  to  employ  the  value 
of  TT  to  only  4  or  5  decimal  places.  But  if  the  diameter  is 
large,  and  accuracy  is  required,  it  will  be  necessary  to  employ 
a  corresponding  number  of  decimal  places  of  TT.  The  value  of 
TT  to  ten  decimal  places  is  3.14159,26536, 

and  its  logarithm  is  0.497150. 

Ex.  1.  What  is  the  circumference  of  a  circle  whose  diame- 
ter is  125  feet? 

Ans.,  392.7  feet. 

Ex.  2.  If  the  diameter  of  the  earth  is  7912  miles,  what  is 
its  circumference  ? 

Ans.,  24856.28  miles. 

Ex.  3.  If  the  diameter  of  the  earth's  orbit  is  189,761,000 
miles,  what  is  its  circumference  ? 

Ans.,  596,151,764  miles. 

To  obtain  this  answer,  the  value  of  TT  must  be  taken  to  al 
least  eight  decimal  places. 

PROBLEM  VII. 

(94.)  To  find  the  diameter  of  a  circle  from  its  circum 
ference. 

RULE  I. 

Divide  the  circumference  by  3.14159. 

This  rule  is  an  obvious  consequence  from  the  preceding 
To  divide  by  a  number  is  the  same  as  to  multiply  by  its  re- 
ciprocal ;  and,  since  multiplication  is  more  easily  performed 
than  division,  it  is  generally  most  convenient  to  multiply  by 
the  reciprocal  of  TT,  which  is  0.3183099.  Hence  we  have 

RULE  II. 
Multiply  the  circumference  by  0.31831. 


MENSURATION   OF    SURFACES.  67 

Ex.  1.  What  is  the  diameter  of  a  circle  whose  circumference 
is  875  feet  ? 

Ans.,  278.52  feet. 

Ex.  2.  If  the  circumference  of  the  moon  is  6786  miles,  what 
is  its  diameter? 

Ans.,  2160  miles. 

Ex.  3.  If  the  circumference  of  the  moon's  orbit  is  1,492,987 
iiiles,  what  is  its  diameter  ? 

Ans.,  475,233  miles 

PROBLEM  VIII. 
(95.)  To  find  the  length  of  an  arc  of  a  circle. 

RULE  I. 

As  360  is  to  the  number  of  degrees  in  the  arc,  so  is  the  civ  - 
cumference  of  the  circle  to  the  length  of  the  arc. 

This  rule  follows  from  Prop.  14,  B.  III.,  in  Geometry,  where 
it  is  proved  that  angles  at  the  center  of  a  circle  have  the  same 
ratio  with  the  intercepted  arcs. 

Ex.  1.  "What  is  the  length  of  an  arc  of  22°,  in  a  circle  whose 
diameter  is  125  feet? 

The  circumference  of  the  circle  is  found  to  be  392.7  feet. 

Then  360  :  22  : :  392.7  :  23.998  feet. 
Ex.  2.  If  the  circumference  of  the  earth  is  24,856.28  miles 
what  is  the  length  of  one  degree  ? 

Ans.,  69.045  miles. 

RULE  II. 

(96.)  Multiply  the  diameter  of  the  circle  by  the  number  of 
degrees  in  the  arc,  and  this  product  by  0.0087266. 

Since  the  circumference  of  a  circle  whose  diameter  is  unity 
is  3.14159,  if  we  divide  this  number  by  360,  we  shall  obtain 
the  length  of  an  arc  of  one  degree,  viz.,  0.0087266.  If  we 
multiply  this  decimal  by  the  number  of  degrees  in  any  arc,  we 
shall  'obtain  the  length  of  that  arc  in  a  circle  whose  diameter 
is  unity  ;  and  this  product,  multiplied  by  the  diameter  of  any 
other  circle,  will  give  the  length  of  an  arc  of  the  given  num- 
her  of  degrees  in  that  circle. 


T  11 1  G  0  N  0  M  E  T  R  Y, 


Ex.  3.  What  is  the  length  of  an  arc  of  25°,  in  a  circle  whose 
radius  is  44  rods  ? 

Ans.,  19.198  rods. 

Ex.  4.  What  is  the  length  of  an  arc  of  11°  15',  in  a  circle 
whose  diameter  is  1234  feet  ? 

Ans.,  121.147  feet. 

(97.)  If  the  number  of  degrees  in  an  arc  is  not  given,  it  may 
be  computed  from  the  radius  of  the  circle, 
and  either  the  chord  or  height  of  the  arc. 
Thus,  let  AB  be  the  chord,  and  DE  the 
height  of  the  arc  ADB,  and  C  the  center 
of  the  circle.  Then,  in  the  right-angled  tri- 
angle ACE, 

(  AE  :  sin. 


(  CE  :  cos.  ACE, 
either  of  which  proportions  will  give  the  number  of  degrees  in 
half  the  arc. 

If  only  the  chord  and  height  of  the  arc  are  given,  the  diam- 
eter of  the  circle  may  be  found.  For,  by  G-eometry,  Prop.  22, 
Cor.,  B.  IV., 

DE  :  AE  : :  AE  :  EF. 

Ex.  5.  What  is  the  length  of  an  arc  whose  chord  is  6  feet, 
in  a  circle  whoso  radius  is  9  feet  ? 

Ans.,  6.117  feet. 

PROBLEM  IX. 
(98.)  To  find  the  area  of  a  circle. 

RULE  I. 

Multiply  the  circumference  by  half  the  radius. 
For  demonstration,  see  Geometry,  Prop.  12,  B.  VI. 

RULE  II. 

Multiply  the  square  of  the  radius  by  3.14159. 
See  G-eometry,  Prop  13,  Cor.  3,  B.  VI. 
Ex.  1.  What  is  the  area  of  a  circle  whose  diameter  is  18 
feet  ? 

Ans.,  254.469  feet. 


MENSURATION   OF   SURFACES  69 

Ex  2.  What  is  the  area  of  a  circle  whose  circumference  is 
74  feet  ? 

^s.,  435.766  feet. 

Ex.  3.  What  is  the  area  of  a  circle  whose  radius  is  125 
yards  ? 

Ans.,  49087.38  yards. 

PROBLEM  X. 
(99.)  To  find  the  area  of  a  sector  of  a  circle. 

RULE  I 

Multiply  the  arc  of  the  sector  by  half  its  radius. 
See  Geometry,  Prop.  12,  Cor.,  B.  VI. 

RULE  II. 

As  360  is  to  the  number  of  degrees  in  the  arc,  so  is  the  area 
of  the  circle  to  the  area  of  the  sector. 

This  follows  from  Geometry,  Prop.  14,  Cor.  2,  B.  III. 
Ex.  1.  What  is  the  area  of  a  sector  whose  arc  is  22°,  in  a 
circle  whose  diameter  is  125  feet  ? 

The  length  of  the  arc  is  found  to  be  23.998. 
Hence  the  area  of  the  sector  is  749.937. 
Ex.  2.  What  is  the  area  of  a  sector  whose  arc  is  25°,  in  a 
circle  whose  radius  is  44  rods  ? 

Ans.,  422.367  rods. 

Ex.  3.  What  is  the  area  of  a  sector  less  than  a  semicircle, 
whose  chord  is  6  feet,  in  a  circle  whose  radius  is  9  feet  ? 

Ans.,  27.522  feet 

/      PROBLEM  XL 
(100.)  To  find  the  area  of  a  segment  of  a  circle. 

RULE. 

Find  the  area  of  the  sector  which  has  the  same  arc,  and 
also  the  area  of  the  triangle  formed  by  the  chord  of  the  seg- 
ment and  the  radii  of  the  sector. 

Then  take  the  sum  of  these  areas  if  the  segment  is  greater 
than  a  semicircle,  but  take  their  difference  if  it  is  less 


70  TRIGONOMETRY 

It  is  obvious  that  the  segment  AEB  is 
equal  to  the  sum  of  the  sector  ACBE  and 
the  triangle  ACB,  and  that  the  segment   A 
ADB  is  equal  to  the  difference  "between     / 
the  sector  ACBD  and  the  triangle  ACB.       ^ 

Ex.  1.  What  is  the  area  of  a  segment 
whose  arc  contains  280°,  in  a  circle  whose 
diameter  is  50  ? 

The  whole  circle       =       1963.495 
The  sector  =       1527.163 

The  triangle  307.752 

The  segment  =       1834.915,  Ans. 

Ex.  2.  What  is  the  area  of  a  segment  whose  chord  is  20  feel, 
and  height  2  feet  ? 

Ans.,  26.8788  feet. 

Ex.  3.  What  is  the  area  of  a  segment  whose  arc  is  25°,  in 
a  circle  whose  radius  is  44  rods  ? 

Ans. 

(101.)  The  area  of  the  zone  ABHG-,  included  between  two 
parallel  chords,  is  equal  to  the  difference  between  the  segments 
GDI!  and  ADB. 

Ex.  4.  What  is  the  area  of  a  zone,  one  side  of  which  is  96, 
and  the  other  side  60,  and  the  distance  between  them  26  ? 

Ans.,  2136.7527. 

The  radius  of  the  circle  in  this  example  will  be  found  to 
be  50. 

PROBLEM  XII. 

(102.)  To  find  the  area  of  a  ring  included  between  the  cir 
cumferences  of  two  concentric  circles. 

RULE. 

Take  the  difference  between  the  areas  of  the  two  circles;  or. 
Subtract  the  square  of  the  less  radius  from  the  square  of  t/u 
greater,  and  multiply  their  difference  by  3.14159. 
For,  according  to  Geometry,  Prop.  13,  Cor.  3,  B.  VI., 
the  area  of  the  greater  circle  is  equal  to  IT  R9, 
and  the  area  of  the  smaller,  TT  r*. 

Their  difference,  or  the  area  of  the  ring,  b  IT  (R2— r2). 


MENSURATION   OF    SOLUS.  71 

Ex.  1.  The  diameters  of  two  concentric  circles  are  60  and 
50.  What  is  the  area  of  the  ring  included  between  their  cir- 
cumferences ? 

Ans.,  863.938. 

Ex.  2.  The  diameters  of  two  concentric  circles  are  320  ami 
280  "What  is  the  area  of  the  ring  included  between  their  cir- 
cumferences ? 

Ans.,  18849.55 

PROBLEM  XIII. 
(103.)  To  find  the  area  of  an  ellipse. 

RULE. 

Multiply  the  product  of  the  semi-axes  by  3.14159. 
For  demonstration,  see  Geometry,  Ellipse,  Prop.  21. 
Ex.  1.  What  is  the  area  of  an  ellipse  whose  major  axis  is 
70  feet,  and  minor  axis  60  feet  ? 

Ans.,  3298.67  feet. 

Ex.  2.  What  is  the  area  of  an  ellipse  whose  axes  are  340 
and  310? 

Ans.,  82780.896 

PROBLEM  XIV. 
(104.)  To  find  the  area  of  a  parabola. 

RULE. 

Multiply  the  base  by  two  thirds  of  the  height. 
For  demonstration,  see  Geometry,  Parabola,  Prop.  12. 
Ex.  1.  What  is  the  area  of  a  parabola  whose  base  is  18  feet, 
and  height  5  feet  ? 

Ans.,  60  feet. 

Ex.  2.  What  is  the  area  of  a  parabola  whose  base  is  525 
feet,  and  height  350  feet  ? 

Ans.,  122500  feet 

'    MENSURATION  UF  SOLIDS. 

(105.)  The  common  measuring  unit  of  solids  is  a  cube, 
whose  faces  are  squares  of  the  same  name ;  as,  a  cubic  inch. 
a  cubic  foot,  &c.  This  measuring  unit  is  not,  however,  of 


72  TRIGONOMETRY. 

necessity  a  cube  whose  faces  are  squares  of  the  same  name, 
Thus  a  "bushel  may  have  the  form  of  a  cube,  but  its  faces  can 
only  be  expressed  by  means  of  some  unit  of  a  different  denom- 
ination. The  following  is 

The  Table  of  Solid  Measure. 
1728     cubic  inches  =  1  cubic  foot. 
27     cubic  feet       =   1  cubic  yard. 
4492J   cubic  feet       =   1  cubic  rod. 
231     cubic  inches   =   1  gallon  (liquid  measure). 
268.8  cubic  inches  =  1  gallon  (dry  measure). 
2150.4  cubic  inches  =   1  bushel. 

PROBLEM  I. 
(106.)  To  find  the  surface  of  a  right  prism. 

RULE. 

Multiply  the  perimeter  of  the  base  by  the  altitude  for  the 
convex  surface.     To  this  add  the  areas  of  the  two  ends  when 
the  entire  surface  is  required. 
See  Geometry,  Prop.  1,  B.  VIII. 

Ex.  1.  What  is  the  entire  surface  of  a  parallelepiped  whose 
altitude  is  20  feet,  breadth  4  feet,  and  depth  2  feet  ? 

Ans.,  256  square  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  pentagonal  prism 
whose  altitude  is  25  feet  6  inches,  and  each  side  of  its  base  3 
feet  9  inches  ? 

Ans.j  526.513  square  feet. 

Ex.  3.  What  is  the  entire  surface  of  an  octagonal  prism 
whose  altitude  is  12  feet  9  inches,  and  each  side  of  its  base  «' 
feet  5  inches  ? 

Ans.j  302.898  square  int. 

PROBLEM  II. 
(107.)  To  find  the  solidity  of  a  prism. 

RULE. 

Multiply  the  area  of  the  base  by  the  altitude. 
See  Geometry,  Prop.  11,  B.  VIII. 


MENSURATION   OF   SOLIDS  73 

Ex.  1.  What  is  the  solidity  of  a  parallelepiped  whose  alti- 
tude is  30  feet,  breadth  6  feet,  and  depth  4  feet  ? 

Ans.,  720  cubic  feet. 

Ex.  2.  What  is  the  solidity  of  a  square  prism  whose  altitude 
is  8  feet  10  inches,  and  each  side  of  its  base  2  feet  3  inches  ? 

Ans.,  44 1 1  cubic  feet. 

Ex.  3  What  is  the  solidity  of  a  pentagonal  prism  whose  a.- 
titude  is  20  feet  6  inches,  and  its  side  2  feet  7  inches  ? 

Ans.,  235.376  cubic  feet. 

PROBLEM  III. 
(108.)  To  find  the  surface  of  a  regular  pyramid. 

RULE. 

•    Multiply  the  perimeter  of  the  base  by  half  the  slant  height 
for  the  convex  surface.     To  this  add  the  area  of  the  bast 
when  the  entire  surface  is  required. 
See  Geometry,  Prop.  14,  B.  VIII. 

Ex.  1.  What  is  the  entire  surface  of  a  triangular  pyramid 
whose  slant  height  is  25  feet,  and  each  side  of  its  base  5  feet? 

Ans.,  198.325  square  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  square  pyramid 
whose  slant  height  is  30  feet,  and  each  side  of  the  base  4  feet  ? 

Ans.,  256  square  feet. 

Ex.  3.  What  is  the  entire  surface  of  a  pentagonal  pyra- 
mid whose  slant  height  is  20  feet,  and  each  side  of  the  base 
3  feet? 

Ans.,  165.484  square  feet. 

PROBLEM  IV. 
(109.)   To  find  the  solidity  of  a  pyramid. 

RULE. 

Multiply  the  area  of  the  base  by  one  third  of  the  altitude. 
See  Geometry,  Prop.  17,  B.  VIII. 

Ex.  1.  What  is  the  solidity  of  a  triangular  pyramid  whose 
altitude  is  25  feet,  and  each  sids  of  its  base  6  feet? 

Ans.,  129.904  cubic  feet 


74  TRIGONOMETRY. 

Ex.  2.  What  is  the  solidity  of  a  square  pyramid  whose  slant 
height  is  22  feet,  and  each  side  of  its  base  10  feet  ? 

Ans.,  714.143  cubic  feet. 

Ex.  3.  What  is  the  solidity  of  a  pentagonal  pyramid  whosu 
altitude  is  20  feet,  and  each  Side  of  its  base  3  feet  ? 

Ans.,  103.228  cubic  feet. 

PROBLEM  V. 

(110.)  To  find  the  surface  of  a  frustum  of  a  regular  pyr- 
amid. 

RULE. 

Multiply  half  the  slant  height  by  the  sum  of  the  perime- 
ters of  the  two  bases  for  the  convex  surface.  To  this  add 
the  areas  of  the  two  bases  when  the  entire  surface  is  re- 
quired. 

See  Geometry,  Prop.  14,  Cor.  1,  B.  VIII. 

Ex.  1.  What  is  the  entire  surface  of  a  frustum  of  a  square 
pyramid  whose  slant  height  is  15  feet,  each  side  of  the  greater 
base  being  4  feet  6  inches,  and  each  side  of  the  less  base  2  feet 
10  inches  ? 

Ans.,  248.278  square  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  frustum  of  an  oc- 
tagonal pyramid  whose  slant  height  is  14  feet,  and  the  sides 
of  the  ends  3  feet  9  inches,  and  2  feet  3  inches  ? 

Ans.,  428.344  square  feet 

PROBLEM  VI. 
(111.)  To  find  the  solidity  of  a  frustum  of  a  pyramid. 

RULE. 

Add  together  the  areas  of  the  two  bases,  and  a  mean  pro- 
portional between  them,  and  multiply  the  sum  by  one  third 
of  the  altitude. 

See  G-eometry,  Prop.  18,  B.  VIII. 

When  the  pyramid  is  regular,  it  is  generally  most  conven- 
ient to  find  the  area  of  its  base  by  Rule  II.,  Art.  92.  If  we 
put  a  to  represent  one  side  of  the  lower  base,  and  b  one  side 
of  the  upper  base,  and  +.he  tabular  number  from  Art.  92  b> 


*      M  E  N  S  U  R  A  T  I  O  N     C  F     S  0  L  I  D  S.  75 

T,  the  area  of  the  lower  base  will  be  a2T  ;  thai  of  the  upper 
base  will  be  &2T  ;  and  the  mean  proportional  will  be  al/T. 
Hence,  if  we  represent  the  height  of  the  frustum  by  A,  its  so- 
lidity will  be 


Ex.  1.  What  is  the  solidity  of  a  frustum  of  an  hexagonal 
pyramid  whose  altitude  is  15  feet,  each  side  of  the  greater  end 
being  3  feet,  and  that  of  the  less  end  2  feet  ? 

Ans.,  246.817  cubic  feel. 

Ex.  2.  What  is  the  solidity  of  a  frustum  of  an  octagonal 
pyramid  whose  altitude  is  9  feet,  each  side  of  the  greater  end 
being  30  inches,  and  that  of  the  less  end  20  inches  ? 

Ans.,  191.125  cubic  feet. 

Definition. 

(112.)  A  wedge  is  a  solid  bounded  by  five  planes,  viz.,  a  rec- 
tangular base,  ABCD,  two  trape-  E 

zoids,  ABFE,  DCFE,  meeting  in 
an  edge,  and  two  triangular  ends, 
ADE,  BCF.  The  altitude  of  the 
wedge  is  the  perpendicular  drawn 
from  any  point  in  the  edge  to  the 
plane  of  the  base,  as  EH.  A 

PROBLEM  VII. 
(113.)  To  find  the  solidity  of  a  wedge. 

RULE. 

Add  the  length  of  the  edge  to  twice  the  length  of  the  base, 
and  multiply  the  sum  by  one  sixth  of  the  product  of  the  height 
of  the  luedge  and  the  breadth  of  the  base. 

Demonstration. 

Put  L=AB,  the  length  of  the  base  ; 
"     /=EF,  the  length  of  the  edge  ; 
"     £=BC,  the  breadth  of  the  base  ; 
"     A=EH,  the  altitude  of  the  wedge. 
Now,  if  the  length  of  the  base  is  equal  to  that  of  the  edge, 


?6  TRIGONOMETRY.  • 

it  is  evident  that  the  wedge  is  half  of  a  pri&m  of  the  same  base 
and  height.  If  the  length  of  the  base  is  greater  than  that  of 
the  edge,  let  a  plane,  EGrI,  be  drawn  parallel  to  BCF.  The 
wedge  will  be  divided  into  two  parts,  viz.,  the  pyramid  E  — 
AJGrD,  and  the  triangular  prism  BCF—  Gr. 

The  solidity  of  the  former  is  equal  to  £M(L  —  /),  and  that 
of  the  latter  is  \bhl.     Their  sum  is 


If  the  length  of  the  base  is  less  than  that  of  the  edge,  the 
wedge  will  be  equal  to  the  difference  between  the  prism  and 
pyramid,  and  we  shall  have 

which  is  equal  to 

the  same  result  as  before. 

Ex.  1.  What  is  the  solidity  of  a  wedge  whose  base  is  30 
inches  long  and  5  inches  broad,  its  altitude  12  inches,  and  tho 
length  of  the  edge  2  feet  ? 

Ans.,  840  cubic  inches. 

Ex.  2.  What  is  the  solidity  of  a  wedge  whose  base  is  40 
inches  long  and  7  inches  broad,  its  altitude  18  inches,  and  the 
length  of  the  edge  30  inches  ? 

Ans.,  2310  cubic  inches. 

Definition. 

(114.)  A  rectangular  prismoid  is  a  solid  bounded  by  six 
planes,  of  which  the  two  bases  are  rectangles  having  their  cor- 
responding  sides  parallel,  and  the  four  upright  sides  of  the  sol- 
id are  trapezoids. 

PROBLEM  VIII. 
To  find  the  solidity  of  a  rectangular  prismoid. 

RULE. 

Add  together  the  areas  of  the  two  bases,  and  four  times  tnt 
area  of  a  parallel  section  equally  distant  from  the  bases,  and 
multiply  the  sum  by  one  sixth  of  the  altitude. 

Demonstration. 
Put  L  and  B=  length  and  breadth  of  one  base  ; 


MENSURATION   OF    SOLIDS.  77 

Put  /  and  b  =  length  and  breadth  of  the  other  base; 

"   M    "   m—  length  and  breadth  of  middle  sec.; 

"    h  =the  altitude  of  the  prismoid. 

It  is  evident  that  if  a  plane  be  made  to  pass 
through  the  opposite  edges  of  the  upper  and 
lower  bases,  the  prismoid  will  be  divided  into 
two  wedges,  whose  bases  are  the  bases  of  the 
prismoid,  and  whose  edges  are  L  and  /.  The  solidity  of  these 
wedges,  and,  consequently,  that  of  the  prismoid,  is 


But,  since  M  is  equally  distant  from  L  and  /,  we  have 

2M=L+/,  and  2m=~B+b  ; 
hence         4Mw=(L+/)  (B+b)='BIj+~Bl+bL+bL 

Substituting  4Mm  for  its  value  in  the  preceding  expression, 
we  obtain  for  the  solidity  of  the  prismoid 


Ex.  1.  "What  are  the  contents  of  a  log  of  wood,  in  the  form 
of  a  rectangular  prismoid,  the  length  and  breadth  of  one  end 
oeing  16  inches  and  12  inches,  and  of  the  other  7  inches  and 
4  inches,  the  length  of  the  log  being  24  feet  ? 

Ans.,  16  J  cubic  feet. 

Ex.  2.  What  is  the  solidity  of  a  log  of  hewn  timber,  whoso 
mds  are  18  inches  by  15,  and  14  inches  by  11J,  its  length  be- 
ing 18  feet?  Ans.j  26|i  cubic  feet 

PROBLEM  IX. 

To  compute  the  excavation  or  embankment  for  a  rail-way. 

(115.)  By  the  preceding  rule  may  be  computed  the  amount 
of  excavation  or  embankment  required  in  constructing  a  rail- 
toad  or  canal.  If  we  divide  the  line  of  the  road  into  portions 
10  small  that  each  may  be  regarded  as  a  straight  line,  and 
suppose  an  equal  number  of  transverse  sections  to  be  made, 
the  excavation  or  embankment  between  two  sections  may  be 
regarded  as  a  prismoid,  and  its  contents  found  by  the  pre 
ceding  rule. 

Let  ABCD  represent  the  lower  surface  of  the  supposed  ex- 
cavation, which  we  will  assume  to  be  parallel  to  the  horizon  ; 
and  let  EFGrH  represent  the  upper  surface  of  the  excavation 


73 


TRIGONOMETRY. 


Also,  let  E'A'B'F',  G'C'D'H 

&'      L 


M 


projected  on  a  horizontal  plane, 
represent  the  vertical  sections  at 
the  extremities.  If  we  suppose  ver- 
tical planes  to  pass  through  the  lines 
AC,  BD,  the  middle  part  of  the  ex- 
cavation, or  that  contained  between 
these  vertical  planes,  will  be  a  rect- 
angular prismoid,  of  which  A'B'KI 
will  be  one  base,  and  C'D'ML  the 
other  base.  Its  solidity  will  there- 
fore be  given  by  Art.  114.  The 
parts  upon  each  side  of  the  middle  prismoid  are  also  halves  of 
rectangular  prismoids  ;  or,  if  the  two  parts  are  equal,  they 
may  be  regarded  as  constituting  a  second  prismoid,  one  of 
whose  bases  is  the  sum  of  the  triangles  A'E'I,  B'F'K  ;  and  the 
other  base  is  the  sum  of  the  triangles  C'Gr'L,  D'H'M.  There- 
fore the  volume  of  the  entire  solid  is  equal  to  the  product  of 
one  sixth  of  its  length,  by  the  sum  of  the  areas  of  the  sections 
at  the  two  extremities,  and  four  times  the  area  of  a  parallel 
and  equidistant  section. 

Ex.  1.  Let  ABCDEFGr  represent  the  profile  of  a  tiact  of 


land  selected  for  the  line  of  a  rail- way ;  and  suppose  it  is  re- 
quired, by  cutting  and  embankment,  to  reduce  it  from  its  pres- 
ent hilly  surface  to  one  uniform  slope  from  the  point  A  to 
the  point  Gf. 

The  distance  AH  is  561  feet ;  the  distance  DK  is  320  feet: 

«          "       HI  is  858  feet;     «         "       KL  is  825  feet; 

"          "       ID    is  825  feet;     "         "        LG-  is  3*JO  feet. 
The  perpendicular  BH  is  18  feet;  the  perpendicular  KE  if.  19  feet; 


CI  is 20 feet; 

The  annexed  figure  repre- 
sents a  cross  section,  showing 
the  form  of  the  excavation. 

The  base  of  the  cutting  is  to 


8  feet 


MENSURATION   OF    fee  LIDS.  7f 

be  50  feet  wide,  the  slope  1J  horizontal  to  1  perpendicular; 
that  is,  where  the  depth  ad  is  10  feet,  the  width  of  the  slope 
cd  at  the  surface  will  be  15  feet. 

Calculation  of  the  portion  ABH. 

SSince  BH  is  18  feet,  the  length  of  cd  in  the  cross  section 
will  he  27  feet,  and  cf,  the  breadth  at  the  top  of  the  section, 
will  be  104  feet.  We  accordingly  find,  by  Art.  87,  the  area  of 
the  trapezoid  forming  the  cross  section  at  BH  equal  to 

^2x18=1386  feet. 

4i 

For  the  middle  section,  the  height  is  J  feet,  cd  is  13.5  feet, 
and  cf  is  77  feet.  The  area  of  .the  cross  section  is  therefore 
equal  to 


The  solid  ABH  will  therefore  be  equal  to 

£T/J  -f 

(13864(4x571.5)  -^-=  343332  cubic  feet,  or 
12716  cubic  yards. 

Calculation  of  the  portion  BCIH. 
Since  CI  is  20  feet,  the  length  of  cd  is  30  feet,  and  cf  is 
feet.     The  area  of  the  section  at  CI  is  therefore  equal  to 


For  the  middle  section,  the  height  is  19  feet,  cd  is  28.5 
and  cf  is  107  feet.     The  area  of  the  cross  section  is  therefore 
equal  to 


The  solid  BCIH  will  therefore  be  equal  to 

85S 
(1386  +1600  +(4  X  1491.5)  -—1280136  cubic  feet,  or 

47412.4  cubic  yards. 

Calculation  of  the  portion  CID. 

The  height  of  the  middle  section  is  10  feet  ;  therefore  cf  is 
80  feet,  and  the  area  of  the  cross  section  is 


80  TRIGONOMETRY. 


The  solid  C1D  will  therefore  be  equal  to 

825 
(1600+4X650)  -g-=577500  cubic  feet,  01 

21388.9  cubic  yards. 

The  entire  amount  of  excavation  therefore  is, 
ABH=  12716.0  cubic  yards. 
BCIH-47412.4  " 

CDI=21388.9 
Total  excavation,    81517.3  " 


The  following  is  a  cross  section,  showing  the  fc  rm  }f  the  em- 
bankment. 

The  top  of  the  embankment 
is  to  be  50  feet  wide,  the  slope 
2  to  1  ;  that  is,  where  the  height 
ad  is  10  feet,  the  base  cd  is  to  c 
he  20  feet. 

Calculation  of  the  portion  DKE. 

Since  EK  is  19  feet,  the  length  of  cd  is  38  feet,  and  cf  is 
126  feet.  The  area  of  the  cross  section  at  EK  is  therefore 
equal  to 


For  the  middle  section,  the  height  is  9.5  feet  ;  cd  is  therefore 
19  feet,  and  cf  is  88  feet.  The  area  of  the  cross  section  is 
therefore 

5^X9.5=655.5 

The  solid  DKE  is  therefore  equal  to 

R90 
(1672-HX  655.5)  -^-=  586846.7  cubic-  feet,  or 

21735.1  cubic  yards. 

Calculation  of  the  portion  KEFL. 
Since  LF  is  8  feet,  cd  is  16  feet,  and  cf  is  82  feet, 
area  of  the  section  at  LF  is  therefore  equal  to 


MENSURATION   ^F    SOLIDS.  HI 


The  height  of  the  middle  section  is  13.5  foei  ;  therefore  cd 
is  27  feet,  and  cf  is  104  feet.  The  area  of  the  cros^:  section  in 
therefore  equal  to 

15i+^x  13.5=1039.5. 

«j 

The  solid  KEFL  will  therefore  be  equal  to 

ogr 

(1672+528+4xl039.5)-g-=874225  cubic  feet,  or 

32378.7  cubic  yards. 

* 
Calculation  of  the  portion  LFG-. 

The  height  of  the  middle  section  is  4  feet  ;  therefore  cf  is  60 
*eet,  and  the  area  of  the  cross  section  is  equal  to 

66+50 

—  t,  —  X4=232. 
TS 

The  solid  LFGr  will  therefore  be  equal  to 

OOQ 

(528+4x232)-g-=  =80080  cubic  feet,  or 

2965.9  cubic.  yards. 
The  entire  amount  of  embankment  therefore  is 

DKE  =21735.1  cubic  yards. 
KEFL  =  32378.7  " 

LFG=  2965.9  " 

Total  embankment,  57079.7  " 

T    Ex.  2.  Compute  the  amount  of  excavation  of  the  hill  A  BCD 
from  the  following  data  : 
The  distance  AH  is  325  feet  ;  the  perpendicular  BH  is  12  feet  ; 

"          "       HI  is  672  feet;     "  "  CI  is  13  feet. 

"  "       ID   is  534  feet. 

The  base  of  the  cutting  to  be  50  feet  wide,  and  the  slope  1$ 
horizontal  to  1  perpendicular.          Ans.,  33969  cubic  yards. 

PROBLEM  X. 
(116.)  To  find  the  surface  of  a  regular  polyedron. 

RULE. 

the  area  of  one  of  the  faces  by  the  numbet  oj 
F 


%%  TRIGONOMETRY. 

faces;  or,  Multiply  the  square  of  one  of  the  edges  by  the 
surface  of  a  similar  solid  whose  edge  is  unity. 

Since  all  the  faces  of  a  regular  polyedron  are  equal,  it  is 
evident  that  the  area  of  one  of  them,  multiplied  by  their  num- 
ber, will  give  the  entire  surface.  Also,  regular  solids  of  the 
same  name  are  similar,  and  similar  polygons  are  as  the  squares 
of  their  homologous  sides  (Geom.,  Prop.  26,  B.  IV.).  The  fol- 
lowing table  shows  the  surface  and  solidity  of  regular  poly- 
edrons  whose  edge  is  unity.  The.  surface  is  obtained  by  mul- 
tiplying the  area  of  one  of  the  faces,  as  given  in  Art.  92,  by 
the  number  of  faces.  Thus  the  area  of  an  equilateral  trian- 
gle, whose  side  is  1,  is  0.4330127.  Hence  the  surface  of  a 
regular  tetraedron 

= .4330127  X  4- 1.7320508, 
and  so  on  for  the  other  solids. 

A  Table  of  the  regular  Polyedrons  whose  Edges  are  unity 

Names,  No.  of  Faces.  Surface.  Solidity. 

Tetraedron,  4  1.7320508         0.1178513. 

Hexaedron,  6  6.0000000         1.0000000. 

Octaedron,  8  3.4641016         0.4714045 

Dodecaedron,        12         20.6457288         7.6631189. 
Icosaedron,  20  8.6602540         2.1816950. 

Ex.  1.  What  is  the  surface  of  a  regular  octaedron  whoso 
edges  are  each  8  feet  ? 

Ans.,  221.7025  feet. 

Ex.  2.  What  is  the  surface  of  a  regular  dodecaedron  whose 
fidse  is  12  feet  ? 

Ans.,  2972.985  feet. 

PROBLEM  XI. 
{117.)  To  find  the  solidity  of  a  regular  polyedron. 

RULE. 

Multiply  the  surface  by  one  third  of  the  perpendicular  let 
^11  from  the  center  on  one  of  the  faces;  or,  Multiply  the 
tube  of  one  of  the  edges  by  the  solidity  of  a  similar  polyedron, 
whose  edge  is  unity. 

Since  ths  C.ices  of  a  regular  polyedron  are  similar  and  equal, 


MENSURATION   OF    SOLID s.  83 

and  fche  solid  angles  are  all  equal  to  each  other,  it  is  evident 
that  the  faces  are  all  equally  distant  from  a  point  in  the  solid 
called  the  center.  If  planes  be  made  to  pass  through  the  cen- 
ter and  the  several  edges  of  the  solid,  they  will  divide  it  into 
as  many  equal  pyramids  as  it  has  faces.  The  base  of  each 
pyramid  will  be  one  of  the  faces  of  the  polyedron ;  and  since 
their  altitude  is  the  perpendicular  from  the  center  upon  one  of 
the  faces,  the  solidity  of  the  polyedron  must  be  equal  to  the 
areas  of  all  the  faces,  multiplied  by  one  third  of  this  perpen- 
dicular. 

Also,  similar  pyramids  are  to  each  other  as  the  cubes  ot 
their  homologous  edges  (Geom.^  Prop.  17,  Cor.  3,  B.  YIIL). 
And  since  two  regular  polyedrons  of  the  same  name  may  be 
divided  into  the  same  number  of  similar  pyramids,  they  must 
be  to  each  other  as  the  cubes  of  their  edges. 

(118.)  The  solidity  of  a  tetraedron  whose  edge  is  unity,  may 
he  computed  in  the  following  manner  : 

Let  C— ABD  be  a  tetraedron.  From  one  angle,  C,  let  fall 
a  perpendicular,  CE,  on  the  opposite  face;  C 

draw  EF  perpendicular  to  AD ;  and  join 
OF,  AE.  Then  AEF  is  a  right-angled  tri- 
angle, in  which  EF,  being  the  sine  of  30°, 
is  one  half  of  AE  or  BE  ;  and  therefore 

FE  is  one  third  of  BF  or  CF.     Hence  the     

cosine  of  the  angle  CFE  is  equal  to  \ ;  that  A.  B 

is,  the  angle  of  inclination  of  the  faces  of  the  polyedron  is  70°, 
31'  44".  Also,  in  the  triangle  CAF,  CF  is  the  sine  of  60°, 
which  is  0.866025.  Hence,  in  the  right-angled  triangle  CEF, 
knowing  one  side  and  the  angles,  we  can  compute  C73,  which 
is  found  to  be  0.8164966.  Whence,  knowing  the  base  ABD 
(Art.  92),  we  obtain  the  solidity  of  the  tetraedron  =0.1178513. 

In  a  somewhat  similar  manner  may  the  solidities  of  the 
other  regular  polyedrons,  given  in  Art,  116,  be  obtained. 

Ex.  1.  What  is  the  solidity  of  a  regular  tetraedrou  whose 
edges  are  each  24  inches  ? 

Ans.,  0.9428  feet. 

Ex.  2.  WLat  is  the  sclidity  of  a  regular  icosaedron  whoso 
adges  are  each  20  feet  ? 

Ans.9  17453.56  feet. 


84  T  R  I  G  O  N  0  M  E  T  R  V. 

THE  THREE  ROUND  BODIES. 

PROBLEM  I. 
(119.)  To  find  the  surface  of  a  cylinder. 

RULE. 

Multiply  the  circumference  of  the  base  by  the  altitude  fm 
the,  convex  surface.     To  this  add  the  areas  of  the  two  ends 
when  the  entire  surface  is  required. 
See  Geometry,  Prop.  1,  B.  X. 

Ex.  1.  What  is  the  convex  surface  of  a  cylinder  whose  alti- 
tude is  23  feet,  and  the  diameter  of  its  base  3  feet  ? 

Ans.j  216.77  square  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  cylinder  whose  alti» 
tude  is  18  feet,  and  the  diameter  of  its  base  5  feet  ? 

Ans. 

PROBLEM  II. 
(120.)  To  find  the  solidity  of  a  cylinder. 

RULE. 

Multiply  the  area  of  the  base  by  the  altitude. 
See  Geometry,  Prop.  2,  B.  X. 

Ex.  1.  What  is  the  solidity  of  a  cylinder  whose  altitude  is 
IS  feet  4  inches,  and  the  diameter  of  its  base  2  feet  10  inches  ? 

Ans.,  115.5917  cubic  feet. 

Ex.  2.  What  is  the  solidity  of  a  cylinder  whose  altitude  is 
12  feet  11  inches,  and  the  circumference  of  its  base  5  feet  3 
inches  ? 

Ans.,  28.3308  cubic  feet. 

PROBLEM  III. 
(121.)  To  find  the  surface  of  a  cone. 

RULE. 

Multiply  the  circumference  of  the  base  by  half  the  side  for 
the  convex  surface  ;  to  which  add  the  area  of  the  base  when 
the  entire  surface  is  required. 

See  Geometry,  Prop.  3,  B.  X. 


MENSURATION   OF   SOLIDS.  85 

Ex.  1.  "What  is  the  entire  surface  of  a  cone  whose  side  is  10 
feet  and  the  diameter  of  its  base  2  feet  3  inches  ? 

Ans.t  39.319  square  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  cone  whose  side  is  15 
feet,  and  the  circumference  of  its  "base  8  feet  ? 

Ans.)  65.093  square  feet. 

PROBLEM  IV. 
(122.)  To  find  the  solidity  of  a  cone. 

RULE. 

Multiply  the  area  of  the  base  by  one  third  of  the  altitude. 
See  G-eometry,  Prop.  5,  B.  X. 

Ex.  1.  What  is  the  solidity  of  a  cone  whose  altitude  is  12 
feet,  and  the  diameter  of  its  "base  2J  feet  ? 

Ans.,  19.635  cubic  feet. 

Ex.  2.  What  is  the  solidity  of  a  cone  whose  altitude  is  2fi 
feet,  and  the  circumference  of  its  base  6  feet  9  inches  ? 

Ans. 

PROBLEM  V. 
(123.)  To  find  the  surface  of  a  frustum  of  a  cone. 

RULE. 

Multiply  half  the  side  by  the  sum  of  the  circumferences  of 
the  two  bases  for  the  convex  surface;  to  this  add  the  areas 
of  the  two  bases  when  the  entire  surface  is  required. 
See  Geometry,  Prop.  4,  B.  X. 

Ex.  1.  What  is  the  entire  surface  of  a  frustum  of  a  cone, 
the  diameters  of  whose  bases  are  9  feet  and  5  feet,  and  whose 
side  is  16  feet  9  inches  ? 

Ans.,  451.6036  square  feet. 

Ex.  2.  What  is  the  convex  surface  of  a  frustum  of  a  cone 
whose  side  is  10  feet,  and  the  circumferences  of  its  bases  6 
feet  and  4  feet  ? 

Ans.,  50  square  feet. 

PROBLEM  VI. 
(124.)  To  find  the  solidity  of  a  frustum  of  a  cone. 


8(5  TRIGONOMETRY. 

RULE 

Add  together  the  areas  of  the  tivo  bases,  and  a  mean  prc* 
portional  between  them,  and  multiply  the  sum  by  one  third 
of  the  altitude. 

See  Geometry,  Prop.  6,  B.  X. 

If  we  put  R  and  r  for  the  radii  of  the  two  bases,  then  TrR2 
will  represent  the  area  of  one  base,  Trr2  the  area  of  the  other, 
and  vrRr  the  mean  proportional  between  them.  Hence,  if  we 
represent  the  height  of  the  frustum  by  h,  its  solidity  will  be 


Ex.  1.  What  is  the  solidity  of  a  frustum  of  a  cone  whose 
altitude  is  20  feet,  the  diameter  of  the  greater  end  5  feet,  and 
that  of  the  less  end  2  feet  6  inches  ? 

Ans.,  229.074  cubic  feet 

Ex.  2.  The  length  of  a  mast  is  60  feet,  its  diameter  at  the 
greater  end  is  20  inches,  and  at  the  less  end  12  inches  :  what 
is  its  solidity  ?  Ans.,  85.521  cubic  feet 

PROBLEM  VII. 
r     (125.)  To  find  the  surface  of  a  sphere. 

RULE. 

Multiply  the  diameter  by  the  circumference  of  a  great  cir- 
cle ;  or,  Multiply  the  square  of  the  diameter  by  3.14159 

See  Geometry,  Prop.  7,  B.  X. 

Ex.  1.  Required  the  surface  of  the  earth,  its  diameter  be- 
ing 7912  miles.  Ans.,  196,662,896  square  miles. 

Ex.  2.  Required  the  surface  of  the  moon,  its  circumference 
being  6786  miles.  Ans. 

PROBLEM  VIII. 
(126.)  To  find  the  solidity  of  a  sphere. 

RULE. 

Multiply  the  surface  by  one  third  of  the  radius  ;  or,  Mul- 
tiply the  cube  of  the  diameter  by  JTT  ;  that  is,  by  0.5236. 
See  Geometry,  Prop.  8,  B.  X. 
"Where  greal  accuracy  is  required,  the  value  cf  ITT  must  be 


MENSURATION   OF    SOLIDS.  y? 

taken  to  more  than  four  decimal  places.     Its  value,  correct  to 
ten  decimal  places,  is  .52359,87756. 

Ex.  1.  What  is  the  solidity  of  the  earth,  if  it  be  a  sphere 
7912  miles  in  diameter  ? 

Ans.,  259,332,805,350  cubic  miles. 

Ex.  2.  If  the  diameter  of  the  moon  be  2160  miles,  what  is 
its  solidity?  Ans. 

PROBLEM  IX. 
(127.)  To  find  the  surface  of  a  spherical  zone. 

RULE. 

Multiply  the  altitude  of  the  zone  by  the  circumference  of  a 
great  circle  of  the  sphere. 

See  G-eometry,  Prop.  7,  Cor.  1,  B.  X. 

Ex.  1.  If  the  diameter  of  the  earth  be  7912  miles,  what  is 
the  surface  of  the  torrid  zone,  extending  23°  27'  36"  on  each 
side  of  the  equator  ? 

Ans.,  78,293,218  square  miles 

Let  PEP'Q,  represent  a  meridian  of  the  earth;  EQ,  the 
equator  ;  P,  P'  the  poles  ;  AB  one  of  the  P 

tropics,  and  GrH  one  of  the  polar  circles. 
Then  PK  will  represent  the  height  of  one 
of  the  frigid  zones,  KD  the  height  of  one 
of  the  temperate  zones,  and  CD  half  the 
height  of  the  torrid  zone. 

Each  of  the  angles  ACE,  CAD,  and 
G-CK  is  equal  to  23°  27'  36". 

In  the  right-angled  triangle  ACD, 

R : AC  :  :  sin.  CAD  :  CD. 
Also,  in  the    ight-angled  triangle  CGrK, 

R  :  CG-  :  :  cos.  GCK  :  CK, 
Then  PK=PC-KC. 

Where  great  accuracy  is  required,  the  sine  and  cosine  oi 
23°  27'  36"  must  be  taken  to  more  than  six  decimal  places 
The  following  values  are  correct  to  ten  decimal  places : 
Natural  sine  of  23°  27'  36"=. 39810,87431. 
"    cosine  of  23°  27'  36":-. 91733,82302. 


88  TRIGONOMETRY". 

Ex.  2.  If  the  polar  circle  extends  23°  27'  36"  from  the  polo 
find  the  convex  surface  of  either  frigid  zone. 

Ans.,  8,128,252  square  miles. 

Ex.  3.  On  the  same  suppositions,  find  the  surface  of  each'  oi 
the  temperate  zones. 

Ans.,  51,056,587  square  miles. 

PROBLEM  X. 

(128.)  To  find  the  solidity  of  a  spherical  segment  with  one 
base. 

RULE. 

Multiply  half  the  height  of  the  segment  by  the  area  of  the 
base,  and  the  cube  of  the  height  by  .5236,  and  add  the  two 
products. 

See  Geometry,  Prop.  9,  B.  X. 

Ex.  1.  "What  is  the  solidity  of  either  frigid  zone,  supposing 
the  earth  to  be  7912  miles  in  diameter,  the  polar  circles  ex- 
tending 23°  27'  36"  from  the  poles  ? 

Ans.,  1,292,390,176  cubic  miles. 

(129.)  The  solidity  of  a  spherical  segment  of  two  bases  is 
the  difference  between  two  spherical  segments,  each  having  a 
single  base. 

Ex.  2.  On  the  same  supposition  as  in  Ex.  1,  find  the  solid- 
ity of  either  temperate  zone. 

Ans.,  55,032,766,543  cubic  miles 
Ex.  3.  Find  the  solidity  of  the  torrid  zone. 

Ans.,  146,682,491,911  cubic  miles 

PROBLEM  XL 
(130.)  To  find  the  area  of  a  spherical  triangle. 

RULE. 

Compute  the  surface  of  the  quadrantal  triangle,  or  ont 
eighth  of  -the  surface  of  the  sphere.  From  the  sum  of  thf 
three  angles  subtract  two  right  angles  ;  divide  the  remainder 
by  90,  and  multiply  the  quotient  by  the  quadrantal  triangle. 

See  Geometry,  Prop.  20,  B.  IX. 


MENSURATION   OF    SOLIDS.  89 

Ex,  1.  What  is  the  area  of  a  triangle  on  a  sphere  whose  di- 
ameter is  10  feet,  if  the  angles  are  55°,  60°,  and  85°  ? 

Ans.,  8.7266  square  feet. 

Ex.  2.  If  the  angles  of  a  spherical  triangle  measured  on  the 
surface  of  the  earth  are  78°  4'  10",  59°  50'  54",  and  42°  5'  37", 
what  is  the  area  of  the  triangle,  supposing  the  earth  a  sphere, 
of  which  the  diameter  is  7912  miles  ? 

Am.,  3110.794  square  miles. 

If  the  excess  of  the  angles  above  two  right  angles  is  ex- 
pressed in  seconds,  we  must  divide  it  by  90  degrees  also  ex- 
pressed in  seconds ;  that  is,  by  324,000. 

PROBLEM  XII. 
(131.)  To  find  the  area  of  a  spherical  polygon. 

RULE. 

Compute  the  surface  of  the  quadrantal  triangle.  From 
the  sum  of  all  the  angles  subtract  the  product  of  two  right 
angles  by  the  number  of  sides  less  two  ;  divide  the  remainder 
by  90,  and  multiply  the  quotient  by  the  quadrantal  triangle. 

See  Geometry,  Prop.  21,  B.  IX. 

Ex.  1.  "What  is  the  area  of  a  spherical  polygon  of  5  sides  on 
a  sphere  whose  diameter  is  10  feet,  supposing  the  sum  of  the 
angles  to  be  640  degrees  ? 

Ans.j  43.633  square  feet 

62°  33'  13"  ; 


Ex.  2.  The  angles  of  a  spherical 
polygon,  measured  on  the  surface 
of  the  earth,  are 


16'    9"; 
111°  45'    8": 


105°  59'    7" ; 
,  155°  19'  12". 
Required  the  area  of  the  polygon. 

Ans.,  5690.477  square  miles 


BOOK  IV. 

SURVEYING. 

(132.)  THE  term  Surveying  includes  the  measurement  ol 
heights  and  distances,  the  determination  of  the  area  of  portions 
of  the  earth's  surface,  and  their  delineation  upon  paper. 

Since  the  earth  is  spherical,  its  surface  is  not  a  plane  sur- 
face, and  if  large  portions  of  the  earth  are  to  be  measured,  the 
curvature  must  be  taken  into  account ;  but  in  ordinary  sur- 
veying, the  portions  of  the  earth  are  supposed  to  be  so  small 
that  the  curvature  may  be  neglected.  The  parts  surveyed  are 
therefore  regarded  as  plane  figures. 

(133.)  If  a  plummet  be  freely  suspended  by  a  line,  and  al- 
lowed to  come  to  a  state  of  rest,  this  line  is  called  a  vertical 
tine. 

Every  plane  passing  through  a  vertical  line  is  a  vertica. 
plane. 

A  line  perpendicular  to  a  vertical  line  is  a  horizontal  line. 

A  plane  perpendicular  to  a  vertical  line  is  a  horizontal 
plane. 

A  vertical  angle  is  one  the  plane  of  whose  sides  is  vertical. 

A  horizontal  angle  is  one  the  plane  of  whose  sides  is  hori- 
zontal. 

An  angle  of  elevation  is  a  vertical  angle  having  one  side 
horizontal  and  the  other  an  ascending 
line,  as  the  angle  BAD. 

An  angle  of  depression  is  a  vortical 
angle  having  one  side  horizontal  and 
the  other  a  descending  line,  as  the  an- 
gle CDA. 

(134.)  When  distances  are  to  be  found  -A. 
by  trigonometrical  computation,  it  is  necessary  to  measure  at 
least  one  line  upon  the  ground,  and  also  as  many  angles  as 
may  be  necessary  to  render  three  parts  of  every  triangle 
known 


SURVEYING.  91 

.In  the  measurement  of  lines,  the  unit  commonly  employed 
by  surveyors  is  a  chain  four  rods  or  sixty-six  feet  in  length, 
called  Gunter's  Chain,  from  the  name  of  the  inventor.  This 
chain  is  divided  into  100  links.  Sometimes  a  half  chain  \» 
used,  containing  50  links. 

Hence,         1  chain=  100  links    =66  feet; 
1  rod     =    25  links   =  16 \  feet; 
1  link   =7.92  inches  =     ^  of  a  foot  nearly. 

(135.)  To  measure  a  horizontal  line. 

To  mark  the  termination  of  the  chain  in  measuring,  ten  iron 
pins  should  be  provided,  about  a  foot  in  length. 

Let  the  person  who  is  to  go  foremost  in  carrying  the  chain, 
and  who  is  called  the  leader,  take  one  end  of  the  chain  and  the 
ten  pins ;  and  let  another  person  take  the  other  end  of  the 
chain,  and  hold  it  at  the  beginning  of  the  line  to  be  measured. 
When  the  leader  has  advanced  until  the  chain  is  stretched 
tight,  he  must  set  down  one  pin  at  the  end  of  the  chain,  the 
other  person  taking  care  that  the  chain  is  in  the  direction  of 
the  line  to  be  measured.  Then  measure  a  second  chain  in  the 
same  manner,  and  so  on  until  all  the  marking  pins  are  ex- 
hausted.  A  record  should  then  be  made  that  ten  chains  have 
been  meamred,  after  which  the  marking  pins  should  be  re- 
turned to  the  leader,  and  the  measurement  continued  as  be- 
fore until  the  whole  line  has  been  passed  over. 

It  is  generally  agreed  to  refer  all  surfaces  to  a  horizontal 
plane.  Hence,  when  an  inclined  surface,  like  the  side  of  a 
hill,  is  to  be  measured,  the  chain  should  be  maintained  in  a 
horizontal  position.  For  this  purpose,  in  ascending  a  hill,  the 
hind  end  of  the  chain  should  be  raised  from  the  ground  until 
it  is  on  a  level  with  the  fore  end,  and  should  be  held  vertically 
over  the  termination  of  the  preceding  chain.  In  descending  a 
hill,  the  fore  end  of  the  chain  should  be  raised  in  the  same 
manner. 


INSTRUMENTS  FOR  MEASURING  ANGLES. 

In  measuring  angles,  some  instrument  is  used  which  con- 
tains a  portion  of  a  graduated  circle  divided  into  degrees  and 
minutes  These  instruments  may  be  adapted  to  measuring 


>2  TRIGONOMETRY. 

either  horizontal  or  vertical  angles.     The  instrument  most  fre- 
quently employed  for  measuring  horizontal  angles  is  called 

THE  SURVEYOR'S  COMPASS. 

(136.)  The  piincipal  parts  of  this  instrument  are  a  compass- 
box,  a  magnetic  needle,  two  sights,  and  a  stand  for  its  support. 
The  compass-box,  ABC,  is  circular,  generally  about  six  inches 
in  diameter,  and  at  its  center  is  a  small  pin  on  which  the  mag- 
netic needle  is  balanced.  The  circumference  of  the  box  is  di- 
vided into  degrees,  and  sometimes  to  half  degrees  ;  and  the  de- 
grees are  numbered  from  the  extremities  of  a  diameter  both 
ways  to  90°.  The  sights,  DE,  FGr,  are  placed  at  right  angles 


to  the  plane  of  the  graduated  circle,  and  in  each  of  these  there 
is  a  large  and  small  aperture  for  convenience  of  observation 
The  instrument,  when  used,  is  mounted  on  a  tripod,  or  a  single 
staff  pointed  with  iron  at  the  bottom,  so  that  it  may  be  firmly 
placed  in  the  ground. 

Sometimes  two  spirit  levels,  H  and  K,  are  attached,  to  indi- 
cate when  the  plane  of  the  graduated  circle  is  brought  into  a 
horizontal  position. 

(137.)  When  the  magnetic  needle  is  supported  so  as  to  turn 
freely,  and  is  allowed  to  come  to  a  state  of  rest,  the  direction 
it  assumes  is  called  the  magnetic  meridian,  one  end  of  the 
needle  indicating  the  north  point  and  the  other  the  south. 

A  horizontal  lino  perpendicular  to  a  meridian  is  an  east  and 
west  line 


SURVEYING. 


93 


All  the  meridians  passing  through  a  survey  of  moderate  ex- 
tent, are  considered  as  straight  lines  parallel  to  each  other. 

The  bearing  or  course  of  a  line  is  the  angle  which  it  makes 
with  a  meridian  passing  through  one  end ;  and  it  is  reckoned 
from  the  north  or  south  point  of  the  horizon,  toward  the  east 
or  west. 

Thus,  if  NS  represent  a  meridian,  and  the  angle  NAB  is  40", 
then  the  bearing  of  AB  from  the  point  A  is 
40°  to  the  west  of  north,  and  is  written  N.  40° 
W.,  and  read  north  forty  degrees  west. 

The  reverse  bearing  of  a  line  is  the  bearing 
taken  from  the  other  end  of  the  line. 

The  forward  bearing  and  reverse  bearing 
of  a  line  are  equal  angles,  but  lie  between  di- 
rectly opposite  points.  Thus,  if  the  bearing 
of  AB  from  A  is  N.  40°  W.,  the  bearing  of  the 
same  line  from  B  is  S.  40°  JJ. 

(138.)  For  measuring  vertical  angles,  the  instrument  com- 
monly  used  is 

A  QUADRANT. 

It  consists  of  a  quarter  of  a  circle,  usually  made  of  brass, 
and  its  limb,  AB,  is  divided  into 
degrees  and  minutes,  numbered 
from  A  up  to  90°.  It  is  furnish- 
ed either  with  a  pair  of  plain 
sights  or  with  a  telescope,  CD, 
which  is  to  be  directed  toward 
the  object  observed.  A  plumb 
line,  CE,  is  suspended  from  the 
center  of  the  quadrant,  and  in- 
dicates when  the  radius  CB  is 
brought  into  a  vertical  position. 

To  measure  the  angle  of  elevation,  for  example,  of  the  top 
of  a  tower,  point  the  telescope,  CD,  toward  the  tower,  keeping 
the  radius,  CB,  in  a  vertical  position  by  means  of  the  plumb 
line,  CE.  Move  the  telescope  until  the  given  object  is  seen  in 
the  middle  of  the  field  of  view.  The  center  of  the  field  is  in- 
dicated by  two  wires  placed  in  the  focus  of  the  object-glass  of 


94 


TRIGONOMETRY 


the  telescope,  one  wire  being  vertical  and  the  other  horizontal. 
When  the  horizontal  wire  is  made  to  coincide  with  the  sum- 
mit of  the  tower,  the  angle  of  elevation  is  shown  upon  the  arc 
A.B  by  means  of  an  index  which  moves  with  the  telescope. 

As  the  arc  is  not  commonly  divided  into  parts  smaller  than 
half  degrees,  when  great  accuracy  is  required,  some  contriv- 
ance is  needed  for  obtaining  smaller  fractions  of  a  degree. 
This  is  usually  effected  by  a  vernier. 

(139.)  A  Vernier  is  a  scale  of  small  extent,  graduated  in 
such  a  manner  that,  being  moved  by  the  side  of  a  fixed  scale> 
we  are  enabled  to  measure  minute  portions  of  this  scale.  The 
length  of  this  movable  scale  is  equal  to  a  certain  number  of 
parts  of  that  to  be  subdivided,  but  it  is  divided  into  parts  one 
more  or  one  less  than  those  of  the  primary  scale  taken  for  the 
length  of  the  vernier.  Thus,  if  we  wash  to  measure  hundredths 
of  an  inch,  as  in  the  case  of  a  barometer,  we  first  divide  an 
inch  into  ten  equal  parts.  "We  then  construct  a  vernier  equal 
in  length  to  11  of  these  divisions,  but  divide  it  into  10  equal 
parts,  by  which  means  each  division  on  the  vernier  is  Tyth 
longer  than  a  division  of  the  primary  scale. 

Thus,  let  AB  be  the  upper  end  of  a  barometer  tube,  the  mer- 
cury standing  at  the  point  C ;  the  scale  is 
divided  into  inches  and  tenths  of  an  inch, 
and  the  middle  piece,  numbered  from  1 
to  9,  is  the  vernier  that  slides  up  and 
down,  having  10  of  its  divisions  equal  to 
11  divisions  of  the  scale,  that  is,  to  y^-ths 
of  an  inch.  Therefore,  each  division  of 
the  vernier  is  Ty¥ths  of  an  inch ;  or  one 
division  of  the  vernier  exceeds  one  divi- 
sion of  the  scale  by  Tl^th  of  an  inch. 
Now,  as  the  sixth  division  of  the  vernier 
(in  the  figure)  coincides  with  a  division 
of  the  scale,  the  fifth  division  of  the  ver- 
'  nier  will  stand  yl^h  of  an  inch  above  the  nearest  division  of 
the  scale  ;  the  fourth  division  Tf  oths  of  an  inch,  and  the  top 
of  the  vernier  will  be  Tf  wths  of  an  inch  above  the  next  lower 
division  of  the  scale  ;  i.  e.,  the  top  of  the  vernier  coincides  with 
29  66  inches  upon  the  scale.  In  practice,  therefore,  we  ob- 


SURVEYING.  95 

serve  what  division  of  the  vernier  coincides  with  a  division  of 
the  scale  ;  this  will  show  the  hundredths  of  an  inch  to  be  added 
to  the  tenths  next  below  the  vernier  at  the  top. 

A  similar  contrivance  is  applied  to  graduated  circles,  to  ob- 
tain the  value  of  an  arc  with  greater  accuracy.  If  a  circle  is 
graduated  to  half  degrees,  or  30',  and  we  wish  to  measure  sin- 
gle minutes  by  the  vernier,  we  take  an  arc  equal  to  31  divi- 
sions upon  the  limb,  and  divide  it  into  30  equal  parts.  Then 
each  division  of  the  vernier  will  be  equal  to  -J-J-ths  of  a  degree, 
while  each  division  of  the  scale  is  f  £ths  of  a  degree.  That  is, 
each  space  on  the  vernier  exceeds  one  on  the  limb  by  1'. 

In  order,  therefore,  to  read  an  angle  for  any  position  of  the 
vernier,  we  pass  along  the  vernier  until  a  line  is  found  coin- 
ciding with  a  line  of  the  limb.  The  number  of  this  line  from 
the  zero  point  indicates  the  minutes  which  are  to  be  added  to 
the  degrees  and  half  degrees  taken  from  the  graduated  circle. 
Sometimes  a  vernier  is  attached  to  the  common  surveyor's* 
compass. 

(140.)  An  instrument  in  common  use  for  measuring  both 
horizontal  and  vertical  angles  is 

THE  THEODOLITE. 

• 

The  theodolite  has  two  circular  brass  plates,  C  and  D  (see  fig. 
next  page),  the  former  of  which  is  called  the  vernier  plate,  and 
the  latter  the  graduated  limb.  Both  have  a  horizontal  motion 
about  the  vertical  axis,  E.  This  axis  consists  of  two  parts,  one 
external,  and  the  other  internal ;  the  former  secured  to  the 
graduated  limb,  D,  and  the  latter  to  the  vernier  plate,  C,  so  that 
the  vernier  plate  turns  freely  upon  the  lower.  The  edge  of  the 
lower  plate  is  divided  into  degrees  and  half  degrees,  and  this 
is  subdivided  by  a  vernier  attached  to  the  upper  plate  into 
single  minutes.  The  degrees  are  numbered  from  0  to  360. 

The  parallel  plates,  A  and  B,  are  held  together  by  a  ball 
which  rests  in  a  socket.  Four  screws,  three  of  which,  a,  a,  at 
are  shown  in  the  figure,  turn  in  sockets  fixed  to  the  lower  plater 
while  their  heads  press  against  the  under  side  of  the  upper 
plate,  by  which  means  the  instrument  is  leveled  for  observa- 
tion. The  whole  rests  upon  a  tripod,  which  is  firmly  attached 
to  the  body  of  the  instrument. 


T  R  1  Or  O  3V  O  M  E  T  R  Y. 


To  the  vernier  plate,  two  spirit-levels,  c,  c,  are  attached  at 
right  angles  to  each  other,  to  determine  when  the  graduated 
limb  is  horizontal.  A  compass,  also,  is  placed  at  F.  Two 
frames,  one  of  which  is  seen  at  N,  support  the  pivots  of  the 
horizontal  axis  of  the  vertical  semicircle  KL,  on  which  the  tel- 
escope, GrH,  is  placed.  One  side  of  the  vertical  arc  is  divided 
into  degrees  and  half  degrees,  and  it  is  divided  into  single  min- 
utes by  the  aid  of  its  vernier.  The  graduation  commences  at 
the  middle  of  the  arc,  and  reads  both  ways  to  90°.  Under  and 
parallel  to  the  telescope  is  a  spirit-level,  M,  to  show  when  the 
telescope  is  brought  to  a  horizontal  position.  To  enable  us  to 
direct  the  telescope  upon  an  object  with  precision,  two  lines 
called  wires  are  fixed  at  right  angles  to  each  other  in  the  focu* 
of  the  telescope. 

*  To  measure  a  Horizontal  Angle  with  the  Theodolite. 

(141.)  Place  the  instrument  exactly  over  the  station  from 
which  the  angle  is  to  be  measured ;  then  level  the  instrument 
by  means  of  the  screws,  &,  a,  bringing  the  telescope  over  each 
pair  alternately  until  the  two  spirit-levels  on  the  vernier  plate 
retain  their  position,  while  the  instrument  is  turned  entirely 
round  upon  its  axis.  Direct  the  telescope  to  one  of  the  objects 


SURVEYING.  97 

ro  be  observed,  moving  it  until  the  cross-wires  and  Dbject  co 
incide.  Now  read  off  the  degrees  upon  the  graduated  limb, 
and  the  minutes  indicated  \y  the  vernier.  Next,  release  the 
upper  plate  (leaving  the  graduated  limb  undisturbed),  and 
move  it  round  until  the  telescope  is  directed  to  the  second  ob- 
ject, and  make  the  cross-wires  bisect  this  object,  as  was  done 
by  the  first.  Again,  read  off  the  vernier ;  the  difference  be- 
tween this  and  the  former  reading  will  be  the  angle  required. 
The  magnetic  bearing  of  an  object  is  determined  by  simply 
reading  the  angle  pointed  out  by  the  compass-needle  when  the 
object  is  bisected. 

To  measure  an  Angle  of  Elevation  with  the  Theodolite. 

(142.)  Direct  the  telescope  toward  the  given  object  so  that 
it  may  be  bisected  by  the  horizontal  wire,  and  then  read  off 
the  arc  upon  the  vertical  semicircle.  After  observing  the  ob- 
ject with  the  telescope  in  its  natural  position,  it  is  well  to  re- 
volve the  telescope  in  its  supports  until  the  level  comes  upper- 
most, and  repeat  the  observation.  The  mean  of  the  two  meas- 
ures may  be  taken  as  the  angle  of  elevation. 

By  the  aid  of  the  instruments  now  described,  we  may  de- 
termine the  distance  of  an  inaccessible  object,  and  its  height 
above  the  surface  of  the  earth. 

HEIGHTS  AND  DISTANCES. 
PROBLEM  I. 

(143.)  To  determine  the  height  of  a  vertical  object  situated, 
on  a  horizontal  plane. 

Measure  from  the  object  to  any  convenient  distance  in  a 
straight  line,  and  then  take  the  angle  of  elevation  subtended 
by  the  object. 

If  we  measure  the  distance  DE ,  and 
the  angle  of  elevation  ODE,  there  will 
be  given,  in  the  right-angled  triangle 
CDE,  the  base  and  the  angles,  to  find 
the  perpendicular  CE  (Art.  46).  To 
this  we  must  add  the  height  of  the  in- 
strument, to  obtain  the  entire  height 
of  the  object  al'ove  the  plane  AB. 

a 


TRIGONOMETRY 


Ex.  1.   Having  measured  AB  /equal    to  100  feet  from  the 
bottom  of  a  tower  on  a  horizontal 
plane,  I  found  the  angle  of  elevation, 
ODE,  of  the  top  to  be  47°  30',  the 
center    of  the    quadrant  being  five 
feet  above  the  ground.     What  is  the 
height  of  the  tower  ? 
R  :  tang.  CDE  : :  DE  :  CE=109.13. 
To  which  add  five  feet,  and  we  obtain, 
the  height  of  the  tower,  114.13  feet. 

Ex.  2.  From  the  edge  of  a  ditch  18  feet  wide,  surrounding 
a  fort,  the  angle  of  elevation  of  the  wall  was  found  to  be  62° 
40'.  Required  the  height  of  the  wall,  and  the  length  of  a  lad- 
der necessary  to  reach  from  my  station  to  the  top  of  it. 

Ans.  The  height  is  34.82  feet.     Length  of  ladder,  39.20  feet. 

PROBLEM  II. 

(144.)  To  find  the  distance  of  a  vertical  object  whose  height 
is  known. 

Measure  the  angle  of  elevation,  and  we  shall  have  given  the 
angles  and  perpendicular  of  a  right-angled  triangle  to  find  the 
base  (Art.  46). 

Ex.  1.  The  angle  of  elevation  of  the  top  of  a  tower  whose 
height  was  known  to  be  143  feet,  was 
found  to  be  35°.     What  was  its  dis- 
tance ? 

Here  we  have  given  the  angles  of  the 
triangle  ABC,  and  the  side  CB,  to  find 
AB. 

Ans.,  204.22  feet. 

If  the  observer  were  stationed  at  the 
top  of  the  tower  BC,  he  might  find  the  length  of  the  base  AB 
by  measuring  the  angle  of  depression  DC  A,  which  is  equal  to 
BAG. 

Ex.  2.  From  the  top  of  a  ship's  mast,  which  was  80  feet 
above  the  water,  the  angle  of  depression  of  another  ship's  hull 
was  found  to  "be  20°.  What  was  its  distance  ? 

.,  219.80  feet 


SURVEYING. 


99 


*    PROBLEM  III. 

(145.)  To  find  the  height  of  a  vertical  object  standing  on 
an  inclined  plane. 

Measure  the  distance  from  the  object  to  any  convenient  sta- 
tion, and  observe  the  angles  which  the  base-line  makes  with 
lines  drawn  from  its  two  ends  to  the  top  of  the  object. 

If  we  measure  the  base-line  AB,  and  the  two  angles  ABC, 
BAG,  then,  in  the  triangle  ABC, 
we  shall  have  given  one  side  and 
the  angles  to  find  BC. 

Ex.  1.  Wanting  to  know  the 
height  of  a  tower  standing  on  an 
inclined  plane,  BD,  I  measured 
from  the  bottom  of  the  tower  a 
distance,  AB,  equal  to  165  feet ; 
also  the  angle  ABC,  equal  to 
107°  18',  and  the  angle  BAG, 
equal  to  33°  35'.  Required  the 
height  of  the  object. 

sin.  ACB  :  AB  : :  sin.  BAG  :  BC-144.66  feet. 

The  height,  BC,  may  also  be  found  by  measuring  the  dis- 
tances BA,  AD,  and  taking  the  angles  BAG,  BDC.  The  dif- 
ference between  the  angles  BAG  and  BDC  will  be  the  angle 
ACD.  There  will  then  be  given,  in  the  triangle  DAC,  one 
side  and  all  the  angles  to  find  AC  ;  after  which  we  shall  have, 
in  the  triangle  ABC,  two  sides  and  the  included  angle  to  find 
BC. 

Ex.  2.  A  tower  standing  on  the  top  of  a  declivity,  I  meas- 
ured 75  feet  from  its  base,  and  then  took  the  angle  BAG,  47° 
50' ;  going  on  in  the  same  direction  40  feet  further,  I  took  the 
angle  BDC,  38°  30'.  What  was  the  height  of  the  tower  ? 

Ans.,  117.21  feet. 

PROBLEM  IV. 

(146.)  To  find  the  distance  of  an  inaccessible  object, 
Measure  a  horizontal  base-line,  and  also  the  angles  between 
tills  line  and  lines  drawn  from  each  station  to  the  object.     Let 
C  -e  the  object  inaccessible  from  A  and  B.     Then,  if  the  dis* 


100 


TJUGONOME  TRY. 


sin.  C  :  AB 


tance  between  the  stations  A  and  B  be  measured,  as  also  the 
angles  at  A  and  B,  there  will  be  given,  in 
the  triangle  ABC,  the  side  AB  and  the  an- 
gles, to  find  AC  and  BC,  the  distances  of  the 
object  from  the  two  stations. 

Ex.  1.  Being  on  the  side  of  a  river,  and 
wanting  to  know  the  distance  to  a  house 
which  stood  on  the  other  side,  I  measured  A  B 

400  yards  in  a  right  line  by  the  side  of  the  river,  and  found 
that  the  two  angles  at  the  ends  of  this  line,  formed  by  the 
other  end  and  the  house,  were  73°  15'  and  68°  2'.  "What  was 
the  distance  between  each  station  and  the  house  ? 

The  angle  C  is  found  to  be  38°  43'.     Then 

sin.  A  :  BC=612.38; 
sin.  B  :  AC=593.09. 

Ex.  2.  Two  ships  of  war,  wishing  to  ascertain  their  distance 
from  a  fort,  sail  from  each  other  a  distance  of  half  a  mile,  when 
they  find  that  the  angles  formed  between  a  line  from  one  to 
the  other,  and  from  each  to  the  fort,  are  85°  15'  and  83°  45'. 
What  are  the  respective  distances  from  the  fort  ? 

Ans.,  4584.52  and  4596.10  yards. 

PROBLEM  Y. 

(147.)  To  find  the  distance  between  two  objects  separated 
by  an  impassable  barrier. 

Measure  the  distance  from  any  convenient  station  to  each 
of  the  objects,  and  the  angle  included  between  those  lines. 

If  we  wish  to  know  the  distance  between  the  places  C  and 
B,  both  of  which  are  accessible,  but  sep- 
arated from  each  other  by  water,  we  may   c 
measure  the  lines  AC  and  AB,  and  also 
the  angle  A.     We  shall  then  have  given 
two  sides  of  a  triangle  and  the  included 
angle  to  find  the  third  side. 

Ex.  1.  The  passage  between  the  two 
objects  C  and  B  being  obstructed,  I  measured  from  A  to  C  735 
rods,  and  from  A  to  B  840  rods ;  also,  the  angle  A,  equal  to 
55°  40'.  What  is  the  distance  of  the  places  C  and  B  ? 

Ans.,  741.21  rod* 


SURVEYING.  101 

Ex.  2.  In  order  to  find  the  distance  between  two  objects,  C 
•  -d  B,  which  could  not  be  directly  measured,  I  measured 
i  im  C  to  A  652  yards,  and  from  B  to  A  756  yards  ;  also,  the 
angle  A  equal  to  142°  25'.  What  is  the  distance  between  the 
objects  C  and  B  ? 

Ans. 

PROBLEM  VI. 

(148.)  To  find  the  height  of  an  inaccessible  object  above  a 
horizontal  plane. 

First  Method. — Take  two  stations  in  a  vertical  plane  pass- 
ing  through  the  top  of  the  object ;  measure  the  distance  be- 
tween  the  stations  and  the  angle  of  elevation  at  each. 

If  we  measure  the  base  AB,  and  the  angles  DAG,  DBG, 
then,  since  CBA  is  the  sup- 
plement of  DBG,  we  shall 
have,  in  the  triangle  ABG, 
one  side  and  all  the  angles 

to  find  BC.     Then,  in  the  ~  "  ~D 

right-angled  triangle  DBG,  we  shall  have  the  hypothenuse  and 
the  angles  to  find  DC. 

Ex.  1.  What  is  the  perpendicular  height  of  a  hill  whose  an- 
gle of  elevation,  taken  at  the  bottom  of  it,  was  46°  ;  and  100 
yards  farther  off,  on  a  level  with  the  bottom  of  it,  the  angle 
was  31°  ? 

Ans.,  143.14  yards. 

Ex.  2.  The  angle  of  elevation  of  a  spire  I  found  to  be  58°, 
and  going  100  yards  directly  from  it,  found  the  angle  to  be 
only  32°.  What  is  the  height  of  the  spire,  supposing  the  in- 
strument to  have  been  five  feet  above  the  ground  at  each  ob- 
servation ? 

Ans.,  104.18  yards. 

(149.)  Second  Method. — Measure  any  convenient  base-line, 
also  the  angles  between  this  base  and  lines  drawn  from  each 
of  its  extremities  to  the  foot  of  the  object,  and  the  angle  of 
elevation  at  one  of  the  stations. 

Let  DG  be  the  given  object.  If  we  measure  the  horizontal 
base-line  AB,  and  the  angles  CAB,  CBA,  we  can  compute  the 
distance  BC.  Also,  if  we  observe  the  angle  of  elevation  CBD, 


J02 


TRIGON  D  ME  TRY. 


we  shall  have  given,  in  the  right-angled  triangle  BCD,  tke 
base  and  angles  to  find  the  perpen- 
dicular. 

Ex.  1.  Being  on  one  side  of  a  river, 
and  wanting  to  know  the  height  of  a 
spire  on  the  other  side,  I  measured 
500  yards,  AB,  along  the  side  of  the 
river,  and  found  the  angle  ABC =74° 
14',  and  BAC=49°  23' ;  also,  the  an- 
gle of  elevation  CBD=11°  15'.  Required  the  height  of  the 
spire.  Ans.,  271.97  feet. 

Ex.  2.  To  find  the  height  of  an  inaccessible  castle,  I  meas- 
ured a  line  of  73  yards,  and  at  each  end  of  it  took  the  angle  of 
position  of  the  object  and  the  other  end,  and  found  the  one  to  be 
90°,  and  the  other  61°  45' ;  also,  the  elevation  of  the  castle  from 
the  latter  station,  10°  35'.  Required  the  height  of  the  castle. 

Ans.,  86.45  feet 

PROBLEM  VII. 

(150.)  To  find  the  distance  between  two  inaccessible  objects 

Measure  any  convenient  base-line,  and  the  angles  between 
this  base  and  lines  drawn  from  each  of  its  extremities  to  each 
of  the  objects. 

Let  C  and  D  be  the  two  inaccessible  objects.  If  we  meas- 
ure a  base-line,  AB,  and  the  an- 
gles DAB,  DBA,  CAB,  CBA, 
then,  in  the  triangle  DAB,  we 
shall 'have  given  the  side  AB 
and  all  the  angles  to  find  BD ; 
also,  in  the  triangle  ABC,  we 
shall  have  one  side  and  all  the 
angles  to  find  BC ;  and  then,  in 
the  triangle  BCD,  we  shall  have  two  sides,  BD,  BC,  with  the 
included  angle,  to  find  DC. 

Ex.  1.  Wanting  to  know  the  distance  between  a  house  and 
a  mill,  which  were  separated  from  me  by  a  river,  I  measured 
a  base-line,  AB,  300  yards,  and  found  the  angle  CAB =58° 
20',  CAD=37°,  ABD=53°  30',  DBC  =  45°  15'.  ,  What  is  the 
distance  of  the  house  from  the  mill  ?  Ans.,  479.80  yards. 


SURVEYING  103 

Ex.  2.  Wanting  to  know  the  distance  between  two  inaccessi 
ble  objects,  C  and  D,  I  measured  a  base-line,  AB,  28.76  rods, 
and  found  the  angle  CAB=33°,  CAD=66°,  DBA=59°  45', 
and  DBG -76°.  "What  is  the  distance  from  C  to  D  ? 

Ans.,  97.696  rods. 

THE  DETERMINATION  OF  AREAS. 

(151.)  The  area  or  content  of  a  tract  of  land  is  the  horizon- 
tal surface  included  within  its  boundaries. 

When  the  surface  of  the  ground  is  broken  and  uneven,  it  is 
very  difficult  to  ascertain  exactly  its  actual  surface.  Hence 
it  has  been  agreed  to  refer  every  surface  to  a  horizontal  plane  ; 
and  for  this  reason,  in  measuring  the  boundary  lines,  it  is  nec- 
essary to  reduce  them  all  to  horizontal  lines. 

The  measuring  unit  of  surfaces  chiefly  employed  by  survey- 
ors is  the  acre,  or  ten  square  chains. 

One  quarter  of  an  acre  is  called  a  rood. 

Since  a  chain  is  four  rods  in  length,  a  square  chain  contains 
sixteen  square  rods  ;  and  an  acre,  or  ten  square  chains,  con- 
tains 160  square  rods.  Square  rods  are  called  perches.  The 
area  of  a  field  is  usually  expressed  in  acres,  roods,  and  perches, 
designated  by  the  letters  A.,  R.,  P. 

When  the  lengths  of  the  bounding  lines  of  a  field  are  given 
in  chains  and  links,  the  area  is  obtained  in  square  chains  and 
square  links.  Now,  since  a  link  is  T|^  of  a  chain,  a  square 
link  will  be  y£o-XTio-  °f  a  square  chain;  that  is,  y^Jo-o-  °f  a 
chain.  Hence  we  have  the  following 

TABLE. 

1  square  chain= 10,000  square  links. 

1  acre=10  square  chains =100, 000  square  links. 

1  acre=4  roods=160  perches. 

If,  then,  the  linear  dimensions  are  links,  the  area  will  be  ex- 
pressed in  square  links,  and  may  be  reduced  to  square  chains 
by  cutting  off  four  places  of  decimals  ;  if  five  places  be  cut  off, 
the  remaining  figures  will  be  acres.  If  the  decimal  part  of  an 
acre  be  multiplied  by  4,  it  will  give  the  roods,  and  the  result- 
ing  decimal,  multiplied  by  40,  will  give  the  perches. 


104  TRIGONOMETRY. 

(152.)  The  difference  of  latitude,  or  the  northing  or  *outh* 
ing-  of  a  line,  is  the  distance  that  one  end  is  further  north  ol 
south  than  the  other  end. 

Thus,  if  NS  be  a  meridian  passing  through  the  end  A  of  the 
line  AB,  and  BC  be  perpendicular  to  NS,  then  is 
AC  the  difference  of  latitude,  or  northing  of  AB. 

The  departure,  or  the  easting-  or  westing  of  a 
Jne,  is  the  distance  that  one  end  is  further  east  or 
west  than  the  other  end. 

Thus  BC  is  the  departure  or  westing  of  the  line 
AB. 

It  is  evident  that  the  distance,  difference  of  lat- 
.tude,  and  departure  form  a  right-angled  triangle, 
of  which  the  distance  is  the  hypothenuse. 

The  meridian  distance  of  a  point  is  the  perpendicular  let  fall 
from  the  given  point  on  some  assumed  meridian,  and  is  east  or 
west  according  as  this  point  lies  on  the  east  or 'west  side  of  the 
meridian. 

The  meridian  distance  of  a  line  is  the  distance  of  the  middle 
point  of  that  line  from  some  assumed  meridian. 

(153.)  When  a  piece  of  ground  is  to  be  surveyed,  we  begin 
at  one  corner  of  the  field,  and  go  entirely  around  the  field, 
measuring  the  length  of  each  of  the  sides  with  a  chain,  and 
their  bearings  with  a  compass. 

Plotting  a  Survey. 

When  a  field  has  been  surveyed,  it  is  easy  to  draw  a  plan 
of  it  on  paper.  For  this  purpose,  draw  a  line  to  represent  the 
meridian  passing  through  the  first  station  ;  then  lay  off  an  an- 
gle equal  to  the  angle  which  the  first  side  of  the  field  makes 
with  the  meridian,  and  take  the  length  of  the  side  from  a  scale 
of  equal  parts.  Through  the  extremity  of  this  side  draw  a 
second  meridian  parallel  to  the  first,  and  proceed  in  the  same 
manner  with  the  remaining  sides.  This  method  will  be  easily 
understood  from  an  example. 

EXAMPLE  1. 

Draw  a  plan  of  a  field  from  the  following  courses  and  dis- 
tances, as  given  in  the  field-book. 


SURVEYING. 


103 


Stations. 

Bearings. 

Distances. 

1 

N.  45°  E. 

9.30  chains. 

2 

S.  60°  E. 

11.85       « 

3 

S.  20°  W. 

5.30       " 

4 

S.  70°  AY. 

10.90       " 

5 

N.  31°  W. 

9.40       " 

Draw  NS  to  represent  a  meridian  line  ;  in  NS  take  any  con- 
venient point,  as  A,  for  the  first  station,  and  lay  off  an  angle, 
NAB,  equal  to  45°,  the  bear- 
ing  from  A  to  B,  which  will 
give  the  direction  from  A  to 
B.  Then,  from  the  scale  of 
equal  parts,  make  AB  equal 
to  9.30,  the  length  of  the  first  A 
side ;  this  will  give  the  sta- 
tion B.  Through  B  draw  a 
second  meridian  parallel  to 
NS  ;  lay  off  an  angle  of  60°, 
and  make  the  line  BC  equal  to 
11.85.  Proceed  in  the  same 
manner  with  the  other  sides.  If  the  survey  is  correct,  and  tho 
plotting  accurately  performed,  the  end  of  the  last  side,  EA, 
will  fall  on  A,  the  place  of  beginning.  This  plot  is  made  on  a 
scale  of  10  chains  to  an  inch. 

(154.)  To  avoid  the  inconvenience  of  drawing  a  meridian 
through  each  angle  of  the  field,  the  sides  may  be  laid  down 
from  the  angles  which  they  make  with  each  other,  instead  of 
the  angles  which  they  make  with  the  meridian.  Reverse  one 
of  the  bearings,  if  necessary,  so  that  both  bearings  may  run 
from  the  same  angular  point ;  then  the  angle  which  any  twc 
contiguous  sides  make  with  each  other  may  be  determined 
from  the  following 

RULES. 

1.  If  both  courses  are  north  or  south,  and  both  east  or  west, 
subtract  the  less  from  the  greater. 

2.  If  both  are  north  or  south,  but  one  east  and  the  other 
west,  add  them  together. 


106 


TRIGONOMETRY. 


3.  If  one  is  north  and  the  other  south,  hut  hoth  east  or  west, 
subtract  their  sum  from  180°. 

4.  If  one  is  north  and  the  other  south,  one  east  and  the  other 
west,  subtract  their  difference  from  180°. 

Thus  the  angle  CAB  is  equal  to 
^AB-NAC. 

The  angle  CAD  is  equal  to  NAC 


The  angle  DAF  is  equal  to  180° 
-(NAD+SAF). 

The  angle  CAF  is  equal  to  180° 
-(SAF-NAC). 

In  the  preceding  example  we  ac- 
cordingly  find  the  angle 

ABC=105°.  DEA-1010. 

BCD=100°.  EAB=104°. 

CDE=130°. 

"With  these  angles  the  field  may  be  plotted  without  drawing 
parallels. 

EXAMPLE  2. 
The  following  field  notes  are  given  to  protract  the  survey  : 


Stations. 

Bearings. 

Distances. 

1 

N.  50°  30'  E. 

16.50  chains. 

2 

S.  68°  15'  E. 

14.20       « 

3 

S.     9°  45'  E. 

8.45       " 

4 

S.  21°    0'  W. 

6.84       " 

5 

S.  73°  30'  W. 

12.31       « 

6 

N.  78°  15'  W. 

9.76       » 

7 

N.  15°  30'  W. 

11.55       " 

THE  TRAVERSE  TABLE. 

(155.)  The  accompanying  traverse  table  shows  the  difference 
of  latitude  and  the  departure  to  four  decimal  places,  for  dis- 
tances from  1  to  10,  and  for  hearings  from  0°  to  90°,  at  inter- 
vals of  15'.  If  the  hearing  is  less  than  45°,  the  angle  will  he 
found  on  the  left  margin  of  one  of  the  pages  of  the  table,  and 
the  distance  at  the  top  or  bottom  of  the  page ;  the  difference 


SURVEYING.  107 

of  latitude  will  be  found  in  the  column  headed  Lat.  at  the  top 
of  the  page,  and  the  departure  in  the  column  headed  Dep.  If 
the  bearing  is  more  than  45°,  the  angle  will  be  found  on  the 
right  margin,  and  the  difference  of  latitude  will  be  found  in 
the  column  marked  Lat.  at  the  bottom  of  the  page,  and  the 
departure  in  the  other  column.  The  latitudes  and  departures 
for  different  distances  with  the  same  bearing  are  proportional 
to  the  distances.  Therefore  the  distances  may  be  reckoned  as 
tens,  hundreds,  or  thousands,  if  the  place  of  the  decimal  point 
in  each  departure  and  difference  of  latitude  be  changed  ac- 
cordingly. 

Ex.  1.  To  find  the  latitude  and  departure  for  the  course  45'* 
and  the  distance  93. 

Under  distance  9  on  page  141,  and  opposite  45°,  will  be 
found  latitude  6.3640  and  departure  6.3640.  Hence,  for  dis- 
tance 90,  the  latitude  is  63.640,  and  adding  the  latitude  for  the 
distance  3,  viz.,  2.121,  we  find  the  latitude  for  distance  93  to 
be  65.761. 

Ex.  2.  To  find  ^  latitude  and  departure  for  the  course  60° 
and  the  distance  li.c?5. 


The  latitude  for  10  is  5.0000. 
"  "  "  1  is  .5000. 
"  "  "  .8  is  .4000. 
"  «  "  .05  is  .0250. 

Departure  for  10  is    8.6603. 
"          "      1  is      .8660. 
"          "     .8  is      .6928. 
"          "  .05  is      .0433. 

Latitude  for  11.85  is  5.9250. 

Depart,  for  11.85  is  10.2624. 

Ex.  3.  To  find  the  latitude  and  departure  for  the  course 
20°  and  the  distance  5.30. 

Ans.  Latitude  4.98,  and  departure  .1.81. 

The  traverse  table  may  b^  used  not  only  for  obtaining  de- 
parture and  difference  of  latitude,  but  for  finding  by  inspection 
the  sides  and  angles  of  any  right-angled  triangle  ;  for  the  lati- 
tude and  departure  form  the  two  legs  of  a  right-angled  trian- 
gle, of  which  the  distance  is  the  hypothenuse,  and  the  course 
is  one  of  the  acute  angles. 

In  this  manner  we  find  the  latitude  and  departure  for  each 
side  of  the  field  given  in  Example  1,  page  105,  to  be  as  in  the 
following  table : 


108 


TRIGONOMETRY. 


Courses. 

Dis- 
tances. 

Latitude. 

Departure. 

Cor. 
Lat. 

Cor. 
Dep. 

Balanced. 

N. 

S. 

E. 

W. 

N. 

S. 

E. 

W. 

1  N.  45°  E. 
2S.   60°  E. 
3  S.  20°  W. 
4  S.  70°  W. 
5N.  31QW. 

9.30 
11.85 
5.30 
10.90 
9.40 

6.58 
8.06 

5.92 
4.98 
3.73 

6.58 
10.26 

1.81 
10.24 
4.84 

+.01 

+.01 
+.01 
—.01 
—.01 
—.01 

6.58 
8.06 

5.93 
4.98 
3.73 

6.59 
10.27 

1.80 
10.23 
4.83 

Perimeter  46.75 

14.64  14.63 

16.84 

16.89 

14.64 

14.64 

16.86J  16.86 

(156.)  "When  a  field  has  been  correctly  surveyed,  and  the 
latitudes  and  departures  accurately  calculated,  the  sum  of  the 
northings  should  he  equal  to  the  sum  of  the  southings,  and  the 
sum  of  the  eastings  equal  to  the  sum  of  the  westings.  If  the 
northings  do  not  agree  with  the  southings,  and  the  eastings 
with  the  westings,  there  must  he  an  error  either  in  the  survey 
or  in  the  calculation.  In  the  preceding  example,  the  north- 
ings exceed  the  southings  by  one  link,  and  the  westings  ex- 
ceed the  eastings  by  five  links.  Small  errors  of  this  kind  are 
unavoidable ;  but  when  the  error  does  not  exceed  one  link  to 
a  distance  of  three  or  four  chains,  it  is  customary  to  distribute 
the  error  among  the  sides  by  the  following  proportion : 

As  the  perimeter  of  the  field, 

Is  to  the  length  of  one  of  the  sides, 

So  is  the  error  in  latitude  or  departure, 

To  the  correction  corresponding1  to  that  side. 

This  correction,  when  applied  to  a  column  in  which  the  sum 
of  the  numbers  is  too  small,  is  to  be  added;  but  if  the  sum  of 
the  numbers  is  too  great,  it  is  to  be  subtracted. 

We  thus  obtain  the  corrections  in  columns  8  and  9  of  the 
preceding  table  ;  and  applying  these  corrections,  we  obtain  the 
balanced  latitudes  and  departures,  in  which  the  sums  of  the 
northings  and  southings  are  equal,  and  also  those  of  the  east- 
ings  and  westings. 

As  the  computations  are  generally  carried  to  but  two  deci- 
mal places,  the  corrections  of  the  latitudes  and  departures  are 
only  required  to  the  nearest  link,  and  these  corrections  may 
often  be  found  by  mere  inspection  without  stating  a  formal 
proportion.  Thus,  in  the  preceding  example,  since  the  depart- 
ures require  a  correction  of  five  links,  and  the  field  has  five 
sides  which  are  not  very  unequal,  it  is  obvious  that  we  must 
make  a  correction  of  one  link  on  each  side. 


SURVEYING. 


109 


D 


I 


It  is  the  opinion  <^f  some  surveyors  that  when  the  error  in 
latitude  or  departure  exceeds  one  link  for  every  five  chains  of 
the  perimeter,  the  field  should  he  resurveyed ;  but  most  sur- 
veyors do  not  attain  to  this  degree  of  accuracy.  The  error, 
however,  should-  never  exceed  one  link  to  a  distance  of  two  or 
three  chains. 

(157.)  To  find  the  area  of  the  field. 

Let  ABODE  he  the  field 
to  he  measured.  Through 
A,  the  most  western  station, 
draw  the  meridian  NS,  and 
upon  it  let  fall  the  perpen- 
diculars BF,  CG-,  DH,  EL 

Then  the  area  of  the  re- 
quired field  is  equal  to 
FBCDEI-(ABF+AEI). 

But  FBCDEI  is  equal  to 
the  sum  of  the  three  trape- 
zoids  FBCG,  GCDH,  HDEI. 

Also,  if  the  sum  of  the 
parallel  sides  FB,  GC  be  multiplied  by  FG,  it  will  give  twice 
the  area  of  FBCG-  (Art.  87).  The  sum  of  the  sides  G-C,  DH, 
multiplied  by  G-H,  gives  twice  the  area  of  GCDH ;  and  the 
sum  of  HD,  IE,  multiplied  by  HI,  gives  twice  the  area  of 
HDEI. 

Now  BF  is  the  departure  of  the  first  side,  G-C  is  the  sum  of 
the  departures  of  the  first  and  second  sides,  HD  is  the  alge- 
braic sum  of  the  three  preceding  departures,  IE  is  the  algebraic 
sum  of  the  four  preceding  departures.  Then  the  sum  of  the 
parallel  sides  of  the  trapezoids  is  obtained  by  adding  together 
the  preceding  meridian  distances  two  by  two ;  and  if  these 
sums  are  multiplied  by  FG,  GH,  &c.,  which  are  the  corre- 
sponding latitudes,  it  will  give  the  double  areas  of  the  trape- 
zoids. 

(158.)  It  is  most  convenient  to  reduce  all  these  operations 
to  a  tabular  form,  according  to  the  following 

RULE, 
Having'  arranged  the  balanced  latitudes  and  departures  in 


110 


TRIGONOMETRY. 


their  appropriate  columns,  draw  a  meridian  through  the,  most 
eastern  or  western  station  of  the  survey,  and,  calling  this  tht. 
first  station,  form  a  column  of  double  meridian  distances. 

The  double  meridian  distance  of  the  first  side  is  equal  to 
its  departure ;  and  the  double  meridian  distance  of  any  side 
is  equal  to  the  double  meridian  distance  of  the  preceding  side, 
plus  its  departure,  plus  the  departure  of  the  side  itself. 

Multiply  each  double  meridian  distance  by  its  correspond- 
ing northing  or  southing,  and  place  the  product  in  the  column 
of  north  or  south  areas.  The  difference  between  the  sum  of 
the  north  areas  and  the  sum  of  the  south  areas  will  be  double 
'the  area  of  the  field. 

It  must  be  borne  in  mind  that  by  the  term  plus  in  this  rule 
is  to  be  understood  the  algebraic  sum.  Hence,  when  the 
double  meridian  distance  and  the  departure  are  both  east  or 
both  west,  they  must  be  added  together  ;  but  if  one  be  east 
and  the  other  west,  the  one  must  be  subtracted  from  the  other. 

The  double  meridian  distance  of  the  last  side  should  always 
be  equal  to  the  departure  for  that  side.  This  coincidence  af- 
fords a  check  against  any  mistake  in  forming  the  column  of 
double  meridian  distances. 

The  preceding  example  will  then  be  completed  as  follows : 


N. 

s. 

E. 

w. 

D.M.D.    |       N.  Areas. 

S.  Areas. 

1 

2 
3 
4 
5 

6.58 
8.06 

5.93 
4.98 
3.73 

6.59 
10.27 

1.80 
10.23 
4.83 

6.59 
23.45 
31.92 
19.89 

4.83 

43.3622 

38.9298 

139.0585 
158.9616 

74.1897 

82.2920 

372.2098 

Twice  the  figure  FBCDEI  is  372.2098  square  chains. 
Twice  the  figure  FBAEI     is    82.2920  " 

The  difference  is     ....  289.9178  " 

Therefore  the  area  of  the  field  is  144.9589  square  chains,  or 

1 4.49589  acres,  which  is  equal  to  14  acres,  1  rood,  39  perches. 
Ex.  2.  It  is  required  to  find  the  contents  of  a  tract  of  land 

of  which  the  following  are  the  field  notes : 


SURVE  Y1NG. 


Ill 


Bta. 
tions. 

Bearings. 

Distances. 

1 

N.  50°  30'  E. 

16.50  chains. 

2 

S.  68°  15'  E. 

14.20      " 

3 

S.     9°  45'  E. 

8.45       * 

4 

S.  21°    0'  W. 

6.84      " 

5 

S.  73°  30'  W. 

12.31      " 

6 

N.  78°  15'  W. 

9.76      « 

7 

N.  17°    0'  W. 

11.64      « 

Calculation. 


Courses. 

Dist. 

Dif.  Lat. 

Departure. 

Cor. 

Balanced. 

D.M. 
D. 

N.  Areas. 

S.  Areas. 

N. 

S. 

K. 

W. 

N. 

s. 

E. 

\v. 

1  N.  50°  3(y  E. 
2  S.  68°  15'  E. 
3S.    9045'E. 
48.21°    O'W. 
5  S.  730  30'  W. 
6N.78°15'W. 
7N.170    O'W. 

16.50 
14.20 
8.45 
6.84 
12.31 
9.76 
11.64 

10.50 

1.99 
11.13 

5.26 
8.33 
6.39 
3.50 

12.73 
13.19 
1.43 

2.45 
11.80 
9.56 
3.40 

.03 
.03 
.01 
.01 
.02 
.02 
.02 

10.47 

1.97 
11.11 

5.29 
8.34 
6.40 
3.52 

12.70 
13.16 
1.42 

2.46 
11.82 
9.58 
3.42 

12.70 
38.56 
53.14 
52.10 
37.82 
16.42 
3.42 

132.9690 

32.3474 
37.9962 

203.9824 
443.1876 
333.4400 
133.1264 

79.70 

23.62  23.48 

27.35 

27.21 

|23.55 

23.55 

27.28 

27.28 

203.3126 

1113.7364 

|  Error  .14 

Error  .14 

203.3126 

Ans.,  45  A.,  2  R.,  3  P. 
Ex.  3.  Required  the  area  of  a 
tract  of  land  of  which  the  follow- 
ing are  the  field  notes  : 


2)910.4238 
455.2119 


Sta- 
tions. 

Bearings. 

Distances. 

1 

N.  58°  45'  E. 

19.84  chains. 

2 

N.  39°  30'  E. 

10.45      « 

3 

S.  45°  15'  E. 

37.26      « 

4 

S.  52°  30  W. 

21.53      « 

5 

S.  34°    0'  E. 

9.12      •« 

6 

S.  66°  15'  W. 

27.69      « 

7 

N.  12°  45'  E. 

24.31      « 

8 

N.  48°  15'  W. 

24.60      « 

Ans.,  130  A.,  2  R.,  23  P, 

Ex.  4.  Required  the  area  of  a  piece  of  land  from  the  follow 
Ing  field  notes : 


112 


TRIGONOMETRY 


r  Stations. 

Bearings. 

Distances. 

1 

N.    5°  15'  E. 

15.17  chains. 

2 

N.  45°  45'  E. 

16.83       « 

3 

N.  32°    0'  W. 

14.26      « 

4 

N.  88°  30'  E. 

19.54      « 

5 

S.  28°  15'  E. 

17.92      " 

6 

S.  40°  45'  W. 

9.71      " 

7 

S.  31°  30'  E.          22.65      " 

8 

S.  14°    0'  W.          18.39      » 

9 

S.  82°  45'  W.         24.80      " 

10 

N.  23°  15'  W.          26.31      " 

Ans.,  173  A.,  0  R.,  23  R 

Ex  5.  Required  the  area  of  a  field  from  the  following  notes 


Stations. 

Bearings. 

Distances. 

1 

N.  32°  15'  E. 

28.74  chains. 

2 

N.  17°  45'  E. 

21.59      « 

3 

S.  81°  30'  E. 

13.38      " 

4 

S.     9°  45'  W. 

11.92      " 

5 

S.  43°    0'  E. 

19.65      « 

6 

N.  25°  30'  E. 

17.26      " 

7 

S.  78°  15'  E. 

18.87      " 

8 

S.     5°  45;  "W. 

31.41      « 

9 

S.  37°  30'  W. 

26.13      « 

10 

N.  69°    0'  W. 

23.86      " 

11 

S.  74°  15'  W. 

20.91      " 

12 

N.  27°  30'  W. 

23.20      " 

Ans.,  304  A.,  2  R.,  9  ?. 
Ex.  6.  Required  the  area  of  a  field  from  the  following  not>s 


Stations. 

Bearings. 

Distances. 

1 

N.  36°  15'  E. 

24.73  chains. 

2 

N.    7°  45'  E. 

11.58      « 

3 

N.  79°  30'  E. 

15.39      « 

4 

S.  86°  45'  E. 

20.56      " 

5 

S.  12°  15'  W. 

18.14      " 

6 

S.  25°    0'  E. 

21.92      " 

7 

S.  58°  30'  W. 

29.27      " 

8 

N.  34°    0'  W. 

19.81      « 

9 

N.  81°  15'  W. 

21.24      " 

Ana..  179  A.,  1  R.,  6 


SURVEYING. 


113 


(159  )  The  field  notes  from  which  the  area  is  to  be  com- 
puted may  be  imperfect.  There  may  be  obstacles  which  pre- 
vent the  measuring  of  one  side,  or  the  notes  may  be  defaced 
BO  as  to  render  some  of  the  numbers  illegible.  If  the  bearings 
and  lengths  of  all  the  sides  of  a  field  except  one  are  given,  the 
remaining  side  may  easily  be  found  by  calculation.  For  the 
difference  between  the  sum  of  the  northings  and  the  sum  of 
the  southings  of  the  given  sides  will  be  the  northing  or  south- 
ing of  the  remaining  side  ;  and  the  difference  between  the  sum 
of  the  eastings  and  the  sum  of  the  ..westings  of  the  given  sides 
will  be  the  easting  or  westing  of  the  remaining  side.  Having, 
then,  the  difference  of  latitude  and  departure  of  the  required 
side,  its  length  and  direction  are  easily  found  by  Trigonome- 
try (Art.  47). 

Ex.  Griven  the  bearings  and  lengths  of  the  sides  of  a  tract 
of  land  as  follows  : 


Stations. 

Bearings. 

Distances. 

1 

N.  18°  15'  E. 

8.93  chains. 

2 

N.  79°  45'  E. 

15.64       " 

3 

S.  25°    0'  E. 

14.27       « 

4 

Unknown. 

Unknown. 

5 

N.  87°  30'  W. 

18.52  chains. 

6 

N.  41°  15'  W. 

12.18       « 

Required  the  bearing  and  distance  of  the  fourth  side. 

Ans.,  S.  15°  33'  E.,  distance  8.62  chains. 
(160.)  There  is  another  method  of  finding  the  area  of  a  field 
which  may  be  practiced  when  great  accuracy  is  not  required 
It  consists  in  first  drawing  a  plan  of  the  field,  as  in  Art.  153 
then  dividing  the  field   into 
triangles   by   diagonal   lines, 
and  measuring  the  bases  and 
perpendiculars  of  the  triangles 
upon  the  same  scale  of  equal 
parts  by  which  the  plot  was 
drawn.     Thas,  if  we  take  Ex. 
1,  and  draw  the  diagonals  AC, 
AD,  the  field  will  be  divided 
into  three  triangles,  whose  area 
a  easily  found  when  we  know 

H 


114 


TRIGONOMETRY. 


the  diagonals  AC,  AD,  and  the  perpendiculars  BF,  DO,  EH. 
The  diagonal  AC  is  found  by  measurement  upon  the  scale  of 
equal  parts  to  be  16.87 ;  the  diagonal  AD  is  15.67  ;  the  perpen- 
licular  BF  is  6.30 ;  DG-  is  4.92 ;  and  EH  is  6.42.  Hence 

the  triangle  ABC=16.87x3.15=  53.14 

"         «       ADC=16.87X2.46=  41.50 

"         "       ADE=15.67x3.21=  50.30 
the  figure  ABODE  ^144.94  sq.  chains. 

This  method  of  finding  the  area  of  a  field  is  very  expedi- 
tious, and  when  the  plot  is  carefully  drawn,  may  afford  results 
sufficiently  precise  for  many  purposes. 

(161.)  To  survey  an  irregular  boundary  by  means  of  off- 
sets. 

WJien  the  boundaries  of  a  field  are  very  irregular,  like  a 
river  or  lake  shore,  it  is  generally  best  to  run  a  straight  line, 
coming  as  near  as  is  convenient  to  the  true  boundary,  and 
measure  the  perpendicular  distances  of  the  prominent  points 
of  the  boundary  from  this  line. 

Let  ABCD  be  a  piece  of  land  to  be  surveyed ;  the  land  be- 
ing bounded  on  the  east  by  a  lake,  and  on  the  west  by  a  creek 
We  select  stations  A,  B,  C, 
D,  so  as  to  form  a  polygon 
which  shall  embrace  most  of 
the  proposed  field,  and  find 
its  area.  "We  then  measure 
perpendiculars  aa',  bb',  cc', 
&c.,  as  also  the  distances  A#, 
ab,  be,  &c.  Then,  consider- 
ing the  spaces  A.aa',  abb' a', 
&c.,  as  triangles  or  trapezoids, 
their  area  may  be  computed ; 
and,  adding  these  areas  to  the 
figure  ABCD,  we  shall  obtain 
the  area  of  the  proposed  field  nearly. 

(162.)  To  determine  the  bearing  and  distance  from  om 
point  to  another  by  means  of  a  series  of  triangles. 

"When  it  is  required  to  find  the  distance  between  two  points 
remote  from  each  oth^r,  we  form  a  series  of  triangles  such  thai 


SURVEYING. 


na 


D 


the  first  and  second  triangles  may  have  one  side  in  common ; 
the  second  and  third,  also,  one  side  in  common ;  the  third  and 
fourth,  &c.  We  then  measure  one  side  of  the  first  triangle 
for  a  base  line^  and  all  the  angles  in  each  of  the  triangles. 
These  dataware  sufficient  to  determine  the  length  of  the  sides 
of  each  triangle  ;  for  in  the  first  triangle  we  have  one  side  and 
the  angles  to  find  the  other  sides.  When  these  are  found,  we 
shall  have  one  side  and  all  the  angles  of  a  second  triangle  to 
find  the  other  sides.  In  the  same  manner  we  may  calculate 
the  dimensions  of  the  third  triangle,  the  fourth,  and  so  on.  We 
shall  illustrate  this  method  by  an  example  taken  from  the 
Coast  Survey  of  the  United  States.  . 

The  object  here  is  to  make  a  survey  of  Chesapeake  Bay  and 
its  vicinity ;  to  determine  with  the 
utmost  precision  the  position  of  the 
most  prominent  points  of  the  country, 
to  which  subordinate  points  may  be 
referred,  and  thus  a  perfect  map  of 
the  country  be  obtained.  According- 
ly, a  level  spot  of  ground  was  select- 
ed on  the  eastern  side  of  the  bay,  on 
Kent  Island,  where  a  base  line,  AB, 
of  more  than  five  miles  in  length, 
was  measured  with  every  precaution. 
A  station,  C,  was  also  selected  upon 
the  other  side  of  the  bay,  near  An- 
napolis, so  situated  that  it  was  visi- 
ble from  A  and  B.  The  three  angles 
of  the  triangle  ABC  were  then  meas- 
ured with  a  large  theodolite,  after 
which  the  length  of  BC  may  be  com- 
puted. A  fourth  station,  D,  is  now  taken  on  the  western  shore 
of  the  bay,  visible  from  C  and  B,  and  all  the  angles  of  the  tri- 
angle BCD  are  measured,  when  the  line  BD  can  be  computed. 
A  fifth  station,  E,  is  now  taken  on  an  island  near  the  eastern 
shore,  visible  both  from  B  and  D,  and  all  the  angles  of  the  tri- 
angle BDE  are  measured,  when  DE  can  be  computed.  Also, 
all  the  angles  of  the  triangle  DEF  are  measured,  and  EF  is 
computed.  Then  all  the  angles  of  the  triangle  EFG-  are  meas- 


11 


116  TRIGONOMETRY. 

ured,  and  FG-  is  computed.  So,  also,  all  the  angles  of  the  tri- 
angle FG-H  are  measured,  and  GrH  is  computed ;  and  thus  a 
chain  of  triangles  may  be  extended  along  the  entire  coast  of 
the  United  States.  To  test  the  accuracy  of  the  work,  it  is 
common  to  measure  a  side  in  one  of  the  triangles  remote  from 
the  first  base,  and  compare  its  measured  length  with  that  de- 
duced by  computation  from  the  entire  series  of  triangles.  This 
line  is  called  a  base  of  verification.  Such  a  base  has  been 
measured  on  Long  Island  ;  and,  indeed,  several  bases  have 
been  measured  on  different  points  of  the  coast.  These  are  all 
3onnected  by  a  triangulation,  and  thus  the  length  of  a  side  in 
any  triangle  may  be  deduced  from  more  than  one  base  line, 
and  the  agreement  of  these  results  is  a  test  of  the  accuracy  of 
the  entire  work.  Thus  the  length  of  one  of  the  sides  of  a  tri- 
angle which  was  twelve  miles,  as  deduced  from  the  Kent  Island 
base,  differed  only  twenty  inches  from  that  derived  from  the 
Long  Island  base,  distant  two  hundred  miles. 

The  superiority  of  this  method  of  surveying  arises  from  the 
circumstance  that  it  is  necessary  to  measure  but  a  small  num- 
ber of  base  lines  along  a  coast  of  a  thousand  or  more  miles  in 
extent ;  and  for  these  the  most  favorable  ground  may  be  se- 
lected any  where  in  the  vicinity  of  the  system  of  triangles. 
All  the  other  quantities  measured  are  angles ;  and  the  pre- 
cision of  these  measurements  is  not  at  all  impaired  by  the  in- 
equalities of  the  surface  of  the  ground.  Indeed,  mountainous 
countries  afford  peculiar  facilities  for  a  trigonometrical  survey, 
since  they  present  heights  of  ground  visible  to  a  great  distance, 
and  thus  permit  the  formation  of  triangles  of  very  large  di- 
mensions. 

(163.)  To  divide  an  irregular  piece  of  land  into  any  two 
given  parts. 

We  first  run  a  line,  by  estimation,  as  near  as  may  be  to  tlu 
required  division  line,  and  compute  the  area  thus  cut  off.  II 
this  is  found  too  large  or  too  small,  we  add  or  subtract  a  tri- 
angle, or  some  other  figure,  as  the  case  may  require.  Sup- 
pose it  is  required  to  divide  the  field  ABODE FGHI  into  two 
equal  parts,  by  a  line  IL,  running  from  the  corner  I  to  the 
opposite  side  CD.  We  first  draw  a  line  from  I  to  D,  and  com- 
pute the  area  of  the  part  DEFGrHI ;  and,  knowing  the  area 


SURVEYING.  11? 

of  the  entire  field,  we  learn  the  area  which  must  be  contained 
in  the  triangle  DIL,  in  order  that 
IL  may  divide  the  field  into  two 
equal  parts.  Having  the  bearings 
and  distances  of  the  sides  DE,  BF, 
&c.,  we  can  compute  the  bearing 
and  distance  of  DI.  Thus  the  an- 
gle IDK  is  known ;  and,  having 
the  hypothenuse  ID,  we  can  com- 
pute the  length  of  the  perpendicu- 
lar IK  let  fall  on  CD.*  Now  the 
base  of  a  triangle  must  be  equal 
to  its  area  divided  by  half  the  al- 
titude. Hence,  if  we  divide  the 
area  of  the  triangle  DIL  by  half  of  IK,  it  will  give  DL. 

In  a  similar  manner  we  might  proceed  if  it  was  required  to 
divide  a  tract  of  land  into  any  two  given  parts. 

Variation  of  the  Needle. 

(164.)  The  line  indicated  by  a  magnetic  needle,  when  free- 
ly supported  and  allowed  to  come  to  a  state  of  rest,  is  called 
the  magnetic  meridian.  This  does  not  generally  coincide  with 
the  astronomical  meridian,  which  is  a  true  north  and  south 
line. 

The  angle  which  the  magnetic  meridian  makes  with  the 
true  meridian  is  called  the  variation  of  the  needle,  and  is  said 
to  be  east  or  west,  according  as  the  north  end  of  the  needle 
points  east  or  west  of  the  north  pole  of  the  earth. 

The  variation  of  the  needle  is  different  in  different  parts  of 
the  earth.  In  some  parts  of  the  United  States  it  is  10°  west, 
and  in  others  10°  east,  while  at  other  places  the  variation  has 
every  intermediate  value.  Even  at  the  same  place,  the  varia- 
tion does  not  remain  constant  for  any  length  of  time.  Hence 
it  is  necessary  frequently  to  determine  the  amount  of  the  varia- 
tion, which  is  easily  done  when  we  know  the  position  of  the 
true  meridian.  The  latter  can  only  .be  determined  by  astro- 
nomical observations.  The  best  method  is  by  observations  of 
the  pole  star.  If  this  star  were  exactly  at  the  pole,  it  would 
always  be  on  the  meridian ;  but,  being  at  a  distance  of  about 


118 


TRIGONOMETRY. 


a  degree  and  a  half  from  the  pole,  it  revolves  about  the  pole  ia 
a  small  circle  in  a  little  less  than  24  hours.  In  about  six  hours 
from  its  passing  the  meridian  above  the  pole,  it  attains  its 
greatest  distance  west  of  the  meridian ;  in  about  six  hours 
more  it  is  on  the  meridian  beneath  the  pole  ;  and  in  about  six 
hours  more  it  attains  its  greatest  distance  east  of  the  meridian. 
If  the  star  can  be  observed  at  the  instant  when  it  is  on  the 
meridian,  either  above  or  below  the  pole,  a  true  north  and  south 
line  may  be  obtained. 

(165.)  The  following  table  shows  the  time  of  the  pole  star'b 
passing  the  meridian  above  the  pole  for, every  fifth  day  of  the 
year: 


1st  Day. 

6th  Day. 

llth  Day. 

16th  Day. 

21st  Day. 

26th  Day. 

January  .  .  . 
February.  .  . 
March  .... 
April  .  . 

6  20  P.M. 
4  18     •« 
2  28     " 
0  26     " 

6    0  P.M. 
358     " 
28" 
07" 

5  41  P.M. 
3  39     " 
1  49     « 

11  47  A  M. 

h.    m. 
5  21  P.M. 
3  19     " 
1  29     " 

11  27  A  M 

h.     m. 
5    1  P.M. 
30" 
19" 
11    SAM 

h.    m. 
4  42  P.M. 
2  40     " 
'  0  50     " 

10  48  A  M 

May  
June  

10  28  A.M. 
8  26     " 

10    9  A.M. 
87" 

9  49     " 

7  47     " 

9  29     " 

7  27     " 

99" 
78" 

8  50     " 
6  48     " 

July  . 

6  28     " 

69" 

5  49     " 

5  29     " 

5  10     " 

4  50     " 

August  .... 
September.  . 
October  .  .  . 
November  .  . 
December  .  . 

4  27     " 
2  25     " 
0  26     " 
10  21  P.M. 
823     •< 

47" 
25" 
07" 
10    1  P.M. 
83" 

3  47     " 
1  45     " 
11  43  P.M. 
9  41      « 
7  43     " 

3  27     " 
1  26     " 
11  24  P.M. 
9  22     " 
7  24     " 

38" 
1    6     " 
11    4  P.M. 
92" 
74" 

2  48     " 
0  46     " 
10  44  P.M. 
8  42     " 
6  44     « 

If  the  pole  star  passes  the  meridian  in  the  daytime,  it  can 
not  be  observed  without  a  good  telescope ;  but  llh  58m>  after 
the  dates  in  the  above  table,  the  star  will  be  on  the  meridian 
below  the  pole,  and  during  the  whole  year,  except  in  summer, 
the  pole  star  may  be  seen  with  the  naked  eye  on  the  merid- 
ian either  above  or  below  the  pole.  These  observations  are 
best  made  with  a  theodolite,  but  they  may  be  made  with  a 
common  compass.  At  5h'  59m>  after  the  dates  in  the  above 
table,  the  star  will  have  attained  its  greatest  distance  west  of 
the  meridian  ;  and  511*  59m>  before  these  dates,  it  will  be  at  its 
greatest  distance  east  of  the  meridian.  In  summer,  therefore, 
we  may  observe  the  greatest  eastern  elongation  of  the  pole  star, 
at  which  time  the  star  is  1°  55'  east  of  the  true  meridian  for 
all  places  in  the  neighborhood  of  New  York.  Making  this  al- 
lowance, a  true  meridian  is  easily  obtained ;  after  which,  the 
variation  of  the  needle  is  determined  by  placing  a  compass 
upon  this  line,  turning  the  sights  in  the  same  direction,  and 
noting  the  angle  shown  by  the  needle. 

The  following  table  shows  the  angle  which  the  plane  of  the 


SURVEYING. 


119 


meridian  makes  with  a  vertical  plane  passing  through  the  pole 
star,  when  at  its  greatest  eastern  or  western  elongation,  for 
any  latitude  from  30°  to  44°. 


Lat.  30° 

1°  41' 

Lat.  32° 
1°43' 

Lat.  34° 

1°45' 

Lat.  36° 

1°48' 

Lat.  38° 


Lat.  40°  I  Lat.  42C 
1°  54'     1°  58' 


Lat.  44" 
2°  2' 


(166.)  The  variation  of  the  needle,  in  1840,  for  several  parts 
of  the  United  States,  was  as  follows  : 


Burlington,  Yt.  . 
Boston,  Mass.  . 
Albany,  N.  Y.  . 
New  Haven,  Ct. 
New  York  City 
Philadelphia  .  . 
Washington  City 


9° 
9° 
6° 
6° 
5° 
4° 
1° 


27'  W. 
12' W. 
58'  W. 
13' W. 
34' W. 
8' W. 
20' W. 


Buffalo,  N.  Y.  . 
Cleveland,  Ohio 
Detroit,  Mich.  . 
Charleston,  S.  C. 
Cincinnati,  Ohio 
Mobile,  Ala.  .  . 
St.  Louis,  Mo.  . 


1°  37'  W. 
0°  19'  E. 
1°  56'  E. 
2°  44' 
4°  46' 
7°  5' 
8°  37' 


E. 
E. 
E. 
E. 


Since  1840,  the  variation  in  New  England  has  increased 
about  five  minutes  annually ;  in  New  York  and  Pennsylvania 
it  has  increased  from  three  to  four  minutes  annually.  In  the 
"Western  States  it  decreases  at  about  the  same  rate,  and  in  the 
Southern  States  it  decreases  about  two  minutes  annually. 


LEVELING. 

(167.)  Leveling  is  the  art  of  determining  the  difference  ol 
level  between  two  or  more  places. 

The  surface  of  an  expanse  of  tranquil  water,  or  any  surface 
parallel  to  it,  is  called  a  level  surface.  Points  situated  in  a 
level  surface  are  said  to  be  on  the  same  level,  and  a  line  traced 
on  such  a  surface  is  called  a  line  of  true  level. 

On  account  of  the  globular  figure  of  the  earth,  a  level  sur- 
face is  not  a  plane  surface.  It  is  nearly  spherical ;  and  in  the 
common  operations  of  leveling  it  is  regarded  as  perfectly  so. 
Hence  every  point  of  a  level  surface  is  regarded  as  at  the  same 
distance  from  the  center  of  the  earth ;  and  the  difference  of 
level  of  two  places  is  the  difference  between  their  distances 
from  the  center. 

A  line  of  apparent  level  is  a  straight  line  tangent  to  the  sur- 
face of  the  earth. 

Thus,  if  AB  represent  the  surface  of  the  ocean,  the  two 
places  A  and  B  are  said  to  be  on  the  same  level ;  but  if  AD 


120  TRIGONOMETRY 

be  drawn  tangent  to  the  arc  AB  at  A,  then  AD  is  a  line  o, 
apparent  level.  ^  ^ 

This  is  the  line  which  is  indicated  by  a 
leveling  instrument  placed  at  A.  The  theod- 
olite may  be  employed  for  tracing  horizontal 
lines ;  "but  if  nothing  further  were  required, 
there  would  be  no  occasion  for  graduated  cir- 
cles, and  several  parts  of  the  theodolite  might 
be  dispensed  with.  A  leveling  instrument,  therefore,  usually 
consists  of  a  large  spirit  level  attached  to  a  telescope,  mounted 
upon  a  stand  in  a  manner  similar  to  the  theodolite. 

(168.)  The  surveyor  should  also  be  provided  with  a  pair  of 
leveling  staves.     A  leveling  staff  consists  of  a 
rectangular  bar  of  wood  six  feet  in  length,  di-   Aifi3^^jB 
vided  to  inches  and  sometimes  tenths  of  an 
inch,  and  having  a  groove  running  its  entire 
length.     A  smaller  staff  of  the  same  length, 
called  a  slide,  also  divided  into  inches,  is  in- 
serted in  this  groove,  and  moves  freely  along  it. 

At  the  upper  end  of  the  slide  is  a  rectangu- 
lar hoard  called  a  vane,  AB,  about  six  inches 
wide.  The  vane  is  divided  into  four  equal  parts  by  two  lines, 
one  horizontal  and  the  other  vertical.  Two  opposite  parts  of 
the  vane  are  painted  white,  and  the  other  two  black,  in  order 
that  they  may  be  distinguished  at  a  great  distance. 

To  find  the  difference  of  level  between  any  two  points. 
(169.)  Set  up  the  leveling  staves  perpendicular  to  the  hori- 
zon, and  at  equal  distances  from  the  leveling  instrument 
Having  adjusted  the  level  by  means  of  the  proper  screws,  turn 
the  telescope  to  one  of  the  staves,  and  direct  an  assistant  to 
slide  up  the  vane  until  the  line  AB  coincides  with  the  center 
of  the  telescope,  and  note  the  height  of  this  line  from  the 
ground.  Turn  the  telescope  to  the  other  staff,  and  repeat  the 
same  operation.  Level  in  the  same  manner  from  the  second 
station  to  the  third,  from  the  third  to  the  fourth,  &c.  Then 
the  difference  between  the  sum  of  the  heights  at  the  back  sta- 
tions and  at  the  forward  stations  will  be  equal  to  the  difference 
of  level  between  the  first  station  and  the  last. 


SURVEYING.  121 

If  we  wish,  to  level  fiom  A  to  E,  we  set  up  the  staves  at  a 
convenient  distance, 
AC,  and  midway  be- 
tween them  place  the 
level  B.  Observe 
where  the  line  of  lev- 
el, FGr,  cuts  the  rods,  and  note  the  heights  AF,  CGr.  Their 
difference  is  the  difference  of  level  between  the  first  and  second 
stations.  Take  up  the  level  and  place  it  at  D,  midway  be- 
tween the  rods  C  and  E,  and  observe  where  the  line  of  level, 
HI,  cuts  the  rods,  and  note  the  heights  CH,  El. 

Then          FA— CGr=the  ascent  from  A  to  C, 
and  CH-EI  =the  ascent  from  C  to  E. 

Therefore  (FA+CH)— (CGr+EI)=the  entire  ascent  from 
A  to  E  ;  and  in  the  same  manner  we  may  find  the  difference 
of  level  for  any  distance ;  that  is,  the  difference  between  the 
sum  of  the  heights  at  the  back  stations  and  at  the  forward 
stations  is  equal  to  the  difference  of  level  between  the  first  sta- 
tion and  the  last. 

(170.)  The  following  is  a  copy  of  the  field  notes  for  running 
a  level  from  A  to  E  : 

Back  sights.  Fore  sights. 

Feet.    Inches.  Feet    Inches. 

04  32 

5  10  57 
42  43 
56  12 
4     11  32 
47  13 

6  1  _2 0 

Sum  31       5  .am  20       7 

The  back  sights  being  greater  in  amount  than  the  forward 
sights,  it  is  evident  that  E  is  higher  than  A  by  10  feet  10 
inches. 

The  heights  indicated  by  the  leveling  staves  are  sometimes 
read  off  by  the  assistant,  but  it  is  better  for  the  observer  to 
read  off  the  quantities  himself  through  the  telescope  of  his 
leveling  instrument.  This  may  easily  be  done  provided  the 
graduation  of  the  staff  is  perfectly  listinct ;  and  in  that  case  it 


122 


TRIGONOMETRY. 


is  only  necessary  to  rely  upon  the  assistant  to  hold  the  staff 
perpendicularly.  To  enable  him  to  do  this,  a  small  plummet  is 
suspended. in  a  groove  cut  in  the  side  of  the  staff. 

(171.)  It  must  be  observed  that  the  lines  GrF,  HI  are  lines 
of  apparent  level,  and 
not  of  true  level ;  nev- 
ertheless, we  shall  ob- 
tain the  true  differ- 
ence of  level  between 
A  and  E  by  this  method  if  the  leveling  instrument  is  placed 
midway  between  the  leveling  staves,  because  the  points  Gr  and 
F  will  in  that  case  be  at  equal  distances  from  the  earth's  cen- 
ter. If  the  level  is  not  placed  midway  between  the  staves, 
then  we  must  apply  a  correction  for  the  difference  between 
the  true  and  apparent  level. 

(172.)  To  find  the  difference  between  the  true  and  apparent 
level. 

Let  C  be  the  center  of  the  earth,  AB  a  portion  of  its  surface, 
and  AD  a  tangent  to  the  earth's  surface  at 
A;  then  BD  is  the  difference  between  the 
true  and  apparent  level  for  the  distance  AD. 

Now,  by  G-eom.,  Prop.  11,  B.  IV., 

CD2=AC2+AD2. 
Hence 
and 


CD  =  VAC2+AD2, 


BD  =  </AC2+AD2-BC. 
If  we  put  R=BC,  the  radius  of  the  earth, 


and  A=BD,  the  difference  between  the  true  and  ap- 

parent level,  we  shall  have 

h=  VET+^-R  ; 

that  is,  to  find  the  difference  between  the  true  and  apparent 
level  for  any  distance,  add  the  square  of  the  distance  to  the 
square  of  the  earth's  radius,  extract  the  square  root  of  the. 
sum,  and  subtract  the  radius  of  the  earth. 

If  BD  represent  a  mountain,  or  other  elevated  object,  then 
AD  will  represent  the  distance  at  which  it  can  be  seen  in  con- 
sequence of  the.  curvature  of  the  earth. 


SURVEYING.  123 

Ex.  1.  If  the  diameter  of  the  earth  be  7912  miles,  and  if 
Mount  Mtna,  can  be  seen  at  sea  126  miles,  what  is  its  height  ? 

Arts.,  2  miles. 

Ex.  2.  If  a  straight  line  from  the  summit  of  Chimborazo 
touch  the  surface  of  the  ocean  at  the  distance  of  179  miles, 
what  is  the  height  of  the  mountain  ?  Ans.,  4.05  miles. 

From  the  preceding  formula  we  obtain 


that  is,  d2=2~R/i+k\ 

But  in  the  common  operations  of  leveling,  h  is  very  small  in 
comparison  with  the  radius  of  the  earth,  and  A2  is  very  small 
in  comparison  with  2R^.  If  we  neglect  the  term  7i2,  we  have 


,  , 

whence  =2ft~' 

that  is,  the  difference  between  the  true  and  apparent  level  is 
nearly  equal  to  the  square  of  the  distance  divided  by  the  di- 
ameter of  the  earth. 

Ex.  1.  "What  is  the  difference  between  the  true  and  apparent 
level  for  one  mile,  supposing  the  diameter  of  the  earth  to  be 
7912  miles?  Ans.,  8.008  inches,  or  8  inches  nearly. 

Ex.  2.  "What  is  the  difference  between  the  true  and  apparent 
level  for  half  a  mile  ?  Ans.,  2  inches. 

d? 
In  the  equation  h=-^,  since  2R  is  a  constant  quantity,  h 

A^Jtv 

varies  as  d*  ;  that  is,  the  difference  between  the  true  and  ap- 
parent level  varies  as  the  square  of  the  distance. 

Hence,  the  difference  for  1  mile  being  8  inches,     Ft.  in 
the  difference  for  2  miles  is  8x22=  32  inches^  2  8. 
"  "         3        «       8x32=:  72     "      =  6  0. 

"  "         4        "       8x42=128     "      -10  8. 

"  "        5        "       8x52=200     "     =16  8. 

"  "         6        "       8x62=288     "      -24  0,  &c. 

Topographical  Maps. 

(173.)  It  is  sometimes  required  to  determine  and  represent 
upon  a  map  the  undulations  and  inequalities  in  the  surface  of 


124 


TRIGONOMETRY. 


a  tract  of  land,  Such  a  map  should  give  a  complete  viuw  ol 
the  ground,  so  as  to  afford  the  means  for  an  appropriate  loca- 
tion of  buildings  or  extensive  works.  For  this  purpose,  we 
suppose  the  surface  of  the  ground  to  be  intersected  by  a  num 
ber  of  horizontal  planes,  at  equal  distances  from  each  othei 
The  lines  in  which  these  planes  meet  the  surface  of  the  ground, 
being  transferred  to  paper,  will  indicate  the  variations  in  the 
inclination  of  the  ground ;  for  it  is  obvious  that  the  curves  will 
be  nearer  together  or  further  apart,  according  as  the  ascent  is 
steep  or  gentle. 

Thus,  let  ABCD  be  a  tract  of  broken  ground,  divided  by  a 
stream,  EF,  the  ascent  being  rapid  on  each  bank,  the  ground 
swelling  to  a  hill  A.  E  33 

at  Gr,  and  also  at 
H.  It  is  required 
to  represent  these 
inequalities  upon 
paper,  so  as  to 
give  an  exact  idea 
of  the  face  of  the 
ground.  The  low- 
est point  of  the 
ground  is  at  F. 
Suppose  the  tract 
to  be  intersected  C 
by  a  horizontal  plane  four  feet  above  F,  and  let  this  plane  in- 
tersect the  surface  of  the  ground  in  the  undulating  lines  marked 
4,  one  on  each  side  of  the  stream.  Suppose  a  second  horizon- 
tal plane  to  be  drawn  eight  feet  above  F,  and  let  it  intersect 
the  surface  of  the  ground  in  the  lines  marked  8.  Let  other 
horizontal  planes  be  drawn  at  a  distance  of  12,  16,  20,  24, 
&c.,  feet  above  the  point  F.  The  projection  of  these  lines  of 
le\el  upon  paper  shows  at  a  glance  the  outline  of  the  tract, 
"We  perceive  that  on  the  right  bank  of  the  stream  the  ground 
rises  more  rapidly  on  the  upper  than  on  the  lower  portion  of 
the  map,  as  is  shown  by  the  lines  of  level  being  nearer  to  one 
another.  On  the  right  bank  of  the  stream  the  ascent  is  unin 
terrupted  until  we  reach  G-,  which  is  the  summit  of  the  hill. 
Beyond  G  the  ground  descends  again  toward  B.  On  the  left 


PURVEYING.  \.2b 

oank  of  the  stream  the  ground  rises  to  H ;  but  toward  A  the 
level  line  of  12  feet  divides  into  two  branches,  and  between 
them  the  ground  is  nearly  level. 

(174.)  The  surveys  requisite  for  the  construction  of  such  a 
map  may  be  made  with  a  theodolite  or  common  level. 

The  object  is  to  trace  a  series  of  level  lines  upon  the  surface 
of  the  ground.  For  this  purpose  we  may  select  any  point  on 
the  surface  of  a  hill,  place  the  level  there,  and  run  a  level  line 
around  the  hill,  measuring  the  distances,  and  also  the  angles, 
at  every  change  of  direction.  "We  may  then  select  a  second 
point  at  any  convenient  distance  above  or  below  the  former, 
and  trace  a  second  level  line  around  the  hill,  and  so  on  for  as 
many  curves  as  may  be  thought  necessary.  Such  a  method, 
however,  would  not  always  be  most  convenient  in  practice. 

(175.)  The  following  method  may  sometimes  be  preferable : 
Set  up  the  level  on  the  summit  of  the  hill  at  G-,  and  fix  the  vane 
on  the  leveling  staff  at  an  elevation  of  four  feet  in  addition  to 
the  height  of  the  telescope  above  the  ground.  Then  direct  an 
assistant  to  carry  the  leveling  staff,  holding  it  in  a  vertical  posi- 
tion, toward  K,  till  he  arrives  at  a  point,  as  &,  where  the  vane 
appears  to  coincide  with  the  cross  wires  of  the  telescope. 
This  will  determine  one  point  of  the  curve  line  four  feet  be- 
low Gr.  The  assistant  may  then  proceed  to  the  line  GrB,  and 
afterward  to  GrL,  moving  backward  or  forward  in  each  of  those 
directions  till  he  finds  points,  as  d  and  g-,  at  which  the  vane 
coincides  with  the  cross  wires  of  the  telescope.  The  horizontal 
distance  between  Gr  and  a,  Gr  and  d,  Gr  and  g*,  must  then  be 
measured. 

If  the  leveling  staff  is  sufficiently  long,  the  vane  may  be 
fixed  on  it  at  the  height  of  eight  feet,  in  addition  to  the  height 
of  the  telescope  at  Gr ;  and  the  assistant,  placing  himself  in 
the  directions  GrK,  GrB,  GrL,  must  move  till  the  vane  appears 
to  coincide  with  the  cross  wires  as  before.  The  horizontal  dis- 
tances ab,  de,  g7j,  must  then  be  measured,  and  stakes  driven 
into  the  ground  at  b,  e,  and  h. 

The  level  must  now  be  removed  to  b  ;  and  the  vane  being 
fixed  on  the  staff  at  a  height  equal  to  four  feet,  together  with 
the  height  of  the  instrument  from  the  ground  at  #,  the  as- 
sistant must  proceed  in  the  direction  6K,  and  stop  at  c  whey 


126 


TRIGONOMETRY 


E 


B 


the  vane  coincides  with  the  cross  wires ;  then  tht,  horizontal 
distance  jf  c  from  A. 
b  must  be  meas- 
ured. In  a  like 
manner,  the  op- 
erations may  be 
continued  from  b 
or  c  as  far  as  nec- 
essary toward  K ; 
then,  commenc- 
ing at  e,  and  aft- 
erward at  /*,  they 
may  be  continu- 
ed in  the  same  ~]?  ~D 
way  toward  B  and  L  respectively.  The  angles  which  the  di- 
rections GrK,  GrB,  GrL  make  with  the  magnetic  meridian  being 
found  with  the  compass,  these  directions  may  be  represented 
on  paper.  Then  the  measured  distances  G-a,  ab,  6sc. ;  Gd,  de, 
&c. ;  Gg*,  g*h,  &c.,  being  set  off  on  those  lines  of  direction, 
ourves  drawn  through  a,  d,  g  ;  b,  e,  h  ;  c,  /,  k,  &o.,  will  show 
the  contour  of  the  hill. 

The  map  is  shaded  so  as  to  indicate  the  hills  and  slopes  by 
drawing  fine  lines,  as  in  the  figure,  perpendicular  to  the  hori- 
zontal curves. 

(176.)  Another  method,  which  may  often  be  more  conven- 
ient than  either  of  the  preceding,  is  as  follows :  From  the  sum- 
mit of  the  hill  measure  any  line,  as  GrK,  and  at  convenient 
points  of  this  line  let  stakes  be  driven,  and  their  distances  from 
Gr  be  carefully  measured.  Then  determine  the  difference  of 
level  of  all  these  points  ;  and  if  the  assumed  points  do  not  fall 
upon  the  horizontal  curves  which  are  required  to  be  delineated, 
we  may,  by  supposing  the  slope  to  be  uniform  from  one  stake 
t}  another,  compute  by  a  proportion  the  points  where  the  hori- 
zontal curves  for  intervals  of  four  feet  intersect  the  line  GrK. 
The  same  may  be  done  for  the  lines  GrB  and  GrL,  and  for  other 
lines,  if  they  should  be  thought  necessary. 

(177.)  If  the  surface  of  the  ground  is  gently  undulating,  it 
may  be  more  convenient  to  run  across  the  tract  a  number  of 
lines  parallel  to  one  another.  Drive  stakes  at  each  extremity 


SURVEYING.  127 

M  these  lines,  and  also  at  all  the  points  along  them  where.- 
there  is  any  material  change  in  the  inclination  of  the  ground, 
and  find  the  difference  of  level  between  all  these  stakes,  and 
their  distances  from  each  other.  Then,  if  we  wish  to  draw 
upon  a  map  the  level  lines  at  intervals  of  4,  6,  or  10  feet,  we 
may  compute  in  the  manner  already  explained  the  points 
where  the  horizontal  curves  intersect  each  of  the  parallel  lines. 
The  curve  lines  are  then  to  be  drawn  through  these  points,  ac- 
cording to  the  judgment  of  the  surveyor. 

(178.)  If  it  is  required  to  draw  a  profile  of  the  ground,  i  ^ 
example,  from  Gr  to  K,  draw  a 
straight  line,  Gr'K,  to  represent 
a  horizontal  line  to  which  the 
heights  are  referred,  and  set  off 
G-V,  G'6',  GV,  &c.,  equal  to 
the  distances  of  the  stations  K  c'  ^  ^  G 

from  the  beginning  of  the  line.  At  the  points  G-',  a',  b',  &c., 
erect  perpendiculars,  Gr'Gr,  a'a,  &c.,  and  make  them  equal  to 
the  heights  of  the  respective  stations.  Through  the  tops  of 
these  perpendiculars  draw  the  curved  line  GrK,  and  it  will  be 
the  profile  of  the  hill  in  the  direction  of  the  line  GrK. 

On  setting  out  Rail-iuay  Curves. 

(179.)  It  is  of  course  desirable  that  the  line  of  a  rail-way 
should  be  perfectly  straight  and  horizontal.  This,  however,  is 
seldom  possible  for  any  great  distance ;  and  when  it  becomes 
necessary  to  change  the  direction  of  the  line,  it  should  be  done 
gradually  by  a  curve.  The  curve  almost  universally  employ* 
ed  for  this  purpose  is  the  arc  of  a  circle,  and  such  an  arc  may 
be  traced  upon  the  ground  by  either  of  the  following  methods. 

First  Method. — "When  the  center  of  the  circle  can  be  seen 
from  every  part  of  the  curve. 

Let  AB,  CD  be  two  straight  portions  of  a  road  which  it  is 
desired  to  connect  by  an  arc  of  a  circle.  Set  up  a  theodolite 
at  B  and  another  at  C,  and  from  each  point  range  a  line  at  right 
angles  to  the  lines  AB  and  CD  respectively ;  and  at  the  inter- 
section of  these  lines,  E,  which  will  be  the  center  of  the  circle, 
erect  a  signal  which  can  be  seen  from  any  point  between  B  and 
C.  Produce  the  lines  AB  and  CD  until  they  meet  in  F.  and 


128 


TRIGONOMETRY. 


a.3  a*  F 


on  these  lines  drive  stakes  at  equal  distances,  a,,  a2,  a3,  com- 
mencing  from  the  points  B  and 
C.     If  r  represents  the  radius 
of  the  circle,  and  d  the  distance 
between  the  points  «,,  aa,  &3, 
&o.,  then  (Art.  172), 
VrM^-r 

will  be  the  distance  which  must 
be  set  off  from  the  first  point 
a,,  in  the  direction  fl^E,  to  ob- 
tain a  point  of  the  circular  arc. 
Tn  like  manner, 


will  be  the  distance  to  be  set  off  from  the  point  a2  ,  in  the  di- 
rection #2E  ;  and,  generally, 


will  be  the  distance  to  be  set  off  at  the  nth  points  from  B  and 
C.  For  example,  let  r  be  one  mile,  or  5280  feet,  and  d  equal 
to  100  feet  ;  then, 


V528024-100a-5280=.94  feet, 

will  be  the  distance  albl.     In  a  similar  manner,  we  find  at 
fla,  or  200  feet  from  B,  the  offset  will  be    3.79  feet 


a3,  or  300 
«4,or  400 


35,or500  «  « 

(180.)  Second  Method.— When.  A 
the  center  of  the  circle  can  not  be 
seen  from  every  part  of  the  curve, 
the  offsets  may  be  set  off  perpendic- 
ularly to  the  tangent  BF,  in  which 
case  they  must  be  computed  from 
the  formula 

For,  in  the  annexed  figure, 


8.52  " 
15.13  " 
23.62  " 


EH=  VG-E'-G-H2 , 
that  is,  EH=  Vr*-d*. 

And  afiG=rBH=BE-HE=r-  Vr 


SURVEYING.  12U 

If  r=5280  feet,  we  shall  find  the  offsets  at  intervals  of  100 
feet  to  Le  a,bl^=  .95  feet. 

aab2=  3.79  k; 
a3b3=  8.53  " 
a,b,  -15.17  " 
tf5&5  =23.73  « 

For  small  distances,  the  offsets  will  be  given  with  sufficient 
accuracy  by  the  formula 

<£_ 
2? 
see  Art.  172. 

It  is  very  common  for  surveyors,  after  they  have  found  the 
first  point,  b  l ,  of  the  curve,  to  join  the  points  B,  b , ,  and  produce 
the  line  Bb ,  to  the  distance  d,  and  from  the  end  of  this  line 
set  off  an  offset  to  determine  the  point  b2  ;  then,  producing  the 
line  bllr2J  set  off  a  third  offset  to  determine  the  point  b3,  and 
so  on.  The  objection  to  this  method  is,  that  any  error  com- 
mitted in  setting  out  one  of  the  points  of  the  curve  will  occa- 
sion an  error  in  every  succeeding  one.  "Whenever  this  method, 
therefore,  is  employed,  it  should  be  checked  by  determining  the 
position  of  every  fourth  or  fifth  point  by  independent  compu- 
tation and  measurement. 

(181.)  Third  Method. — Where  the  radius  of  the  curve  is 
rrrmll,  place  a  theodolite  at  B,  and  point  its  telescope  toward 


C  Place  another  theodolite  at  C,  and  point  its  telescope  to- 
ward E,  the  point  of  intersection  of  the  lines  AB,  CD  produced. 
Then,  if  the  former  be  moved  through  any  number  of  degrees 
toward  a^  and  the  latter  the  same  number  of  degrees  toward 
0,,  the  point  al  will  be  a  point  of  the  curve,  for  the  angle 
Ba ,  C  will  be  equal  to  BCD  (Geom.,  Prop.  16,  B.  III.).  In  the 
same  manner,  #2,  #3,  &c.,  any  number  of  points  cf  the  curve, 
may  be  determined.  It  will  be  most  convenient  to  move  the 

J 


130  TRIGONOMETRY. 

theodolites  each  time  through  an  even  number  of  degrees,  for 
example,  an  arc  of  two  degrees,  and  a  stake  must  be  driven  at 
each  of  the  points  of  intersection  &,,  #2,  #3,  &c.  The  ac- 
curacy of  this  method  is  independent  of  any  undulations  in  the 
surface  of  the  ground,  so  that  in  a  hilly  country  this  method 
may  be  preferable  to  any  other. 


When  the  position  of  one  end  of  the  curve  is  not  absolutely 
determined,  the  engineer  may  proceed  more  rapidly.  Suppose 
it  is  required  to  trace  an  arc  of  a  circle  having  a  curvature  of 
two  degrees  ,for  a  hundred  feet. 

Place  a  theodolite  at  C,  the  point  where  the  curve  commen- 
ces, and  lay  off  from  the  line  CE,  toward  B,  an  angle  of  two 
degrees,  and  in  the  direction  of  the  axis  of  the  instrument  set 
.off  a  distance  of  100  feet,  which  will  give  the  first  point  a ,  oi 
the  curve.  Next  lay  off  from  CB  an  angle  of  four  degrees,  and 
from  a  l  set  off  a  distance  of  100  feet,  and  the  point  where  this 
line  cuts  the  axis  of  the  instrument  produced  will  be  the  sec- 
ond point  Q2.  In  the  same  manner,  lay  off  from  CE  an  angle 
of  six  degrees,  ;and  from  #2  set  off  a  distance  of  100  feet,  and 
the  point  where  it  cuts  the  axis  of  the  instrument  produced 
will  be  the  third  point  a  3 .  All  the  points  a  l ,  a2 ,  a 3 ,  etc.,  thus 
determined  lie  in  the  circumference  of  a  circle  (Geom.,  Prop. 
15,  B.  III.).  Circles  thus  drawn  are  generally  made  with  a 
curvature  of  one  or  two  degrees,  or  some  convenient  fraction 
of  a  degree,  for  every  hundred  feet.  This  method  is  very  ex 
tensively  practiced  in  the  United  States. 

Surveying1  Harbors. 

(182.)  In  surveying  a  harbor,  it  is  necessary  to  determine 
the  position  of  the  most  conspicuous  objects,  to  trace  the  out- 
line of  the  shore,  and  discover  the  depth  of  water  in  the  neigh- 
borhood  of  the  channel.  A  smooth,  level  piece  of  ground  is 


SURVEYING. 


131 


chosen,  on  which  a  base  line  of  considerable  length  is  meas- 
ured, and  station  staves  are  fixed  at  its  extremities.  We  also 
erect  station  staves  on  all  the  prominent  points  to  be  surveyed, 
forming  a  series  of  triangles  covering  the  entire  surface  of  the 
harbor.  The  angles  of  these  triangles  are  now  measured  with 
a  theodolite,  and  their  sides  computed.  After  the  principal 
p  ints  have  been  determined,  subordinate  points  may  be  ascer- 
t  ined  by  the  compass  or  plane  table. 

Let  the  following  figure  be  a  map  of  a  harbor  to  be  survey- 


ed. "We  select  the  most  favorable  position  for  a  base  line, 
which  is  found  to  be  on  the  right  of  the  harbor,  from  A  to  B. 
We  also  erect  station  flags  at  the  points  C,  D,  E,  F,  and  Gr. 
Having  carefully  measured  the  base  line  AB,  we  measure  the 
three  angles  of  the  triangle  ABC,  which  enables  us  to  compute 
the  remaining  sides.  We  then  measure  the  three  angles  of  the 
triangle  ACD,  and  by  meaiis  of  the  side  AC,  just  computed, 
we  are  enabled  to  compute  AD  and  CD.  We  then  measure 
the  three  angles  of  the  triangle  CDF,  and  by  means  of  the  side 
CD,  just  found,  we  are  enabled  to  compute  CF  and  DF.  Pro- 
ceeding in  the  same  manner  with  the  triangles  CEF,  DFGr,  we 
are  enabled,  after  measuring  the  angles,  to  compute  the  sides. 
(183.)  Having  determined  the  main  points  of  the  harbor,  we 
nay  proceed  to  a  more  detailed  survey  by  means  of  the  chain 


132 


T  R  I  G  0  X  0  :,I  E  T  R  Y. 


B 


and  compass.  If  it  is  required  to  trace  the  shore,  HCK,  wo. 
commence  at  H,  and  observe  the  hearings  with  the  compas?, 
and  measure  the  distances  with  the  chain.  Where  the  shore 
is  undulating,  it  is  most  convenient  to  run  a  straight  line  foi 
a  considerable  distance,  and  at  frequent  intervals  measure  off- 
sets to  the  shore. 

When  a  great  many  objects  are  to  be  represented  upon  a 
map,  the  most  convenient  instrument  is 

The  Plane  Table. 

(184.)  The  plane  table 
is  a  board  about  sixteen 
inches  square,  designed 
to  receive  a  sheet  of  draw- 
ing paper,  and  has  two 
plates  of  brass  upon  op- 
posite sides,  confined  by 
screws,  for  stretching  and 
retaining  the  paper  upon 
the  board.  The  margin 
of  the  board  is  divided  to 
360  degrees  from  a  cen- 
ter C,  in  the  middle  of 
the  board,  and  these  are 
subdivided  as  minutely  as  the  size  of  the  table  will  admit.  On 
one  side  of  the  board  there  is  usually  a  diagonal  scale  of  equal 
parts.  A  compass  box  is  sometimes  attached,  which  renders 
the  plane  table  capable  of  answering  the  purpose  of  a  survey- 
or's compass. 

The  ruler,  A,  is  made  of  brass,  as  long  as  the  diagonal  of 
the  table,  and  about  two  inches  broad.  A  perpendicular  sight- 
vane,  B,  B,  is  fixed  to  each  extremity  of  the  ruler,  and  the  eye 
looking  through  one  of  them,  the  vertical  thread  in  the  other  is 
made  to  bisect  any  required  distant  object. 

To  the  under  side  of  the  table,  a  center  is  attached  with  a 
ball  and  socket,  or  parallel  plate  screws,  like  those  of  the  the- 
odolite, by  which  it  can  be  placed  upon  a  staff-head ;  and  the 
table  may  be  made  horizontal  by  means  of  a  detached  spirit 
level 


SURVEYING.  1^3 

(165.)  To  prepare  the  table  for  use,  it  mast  be  covered  with 
drawing  paper.  Then  set  up  the  instrument  at  one  of  the 
stations,  for  example,  B  (see  fig.  on  p.  131),  and  fix  a  needle  in 
the  table  at  the  point  on  the  paper  representing  that  station, 
and  place  the  edge  of  the  ruler  against  the  needle.  Then  di- 
rect the  sights  to  the  station  A,  and  by  the  side  of  the  ruler 
draw  a  line  upon  the  paper  to  represent  the  direction  of  AB. 
Then,  with  a  pair  of  dividers,  take  from  the  scale  a  certain 
number  of  equal  parts  to  represent  the  base,  and  lay  off  this 
distance  on  the  base  line.  Having  drawn  the  base  line,  move 
the  ruler  around  the  needle,  direct  the  sights  to  any  object, 
as  L,  and  keeping  it  there,  draw  a  line  along  the  edge  of  the  ru- 
ler. Then  direct  the  sights  in  the  same  manner  to  any  other 
objects  which  are  required  to  be  sketched,  drawing  lines  in  theii 
respective  directions,  taking  care  that  the  table  remains  steady 
during  the  operation. 

Now  remove  the  instrument  to  the  other  extremity  of  the 
base  A,  and  place  the  point  of  the  paper  corresponding  to  that 
extremity  directly  over  it.  Place  the  edge  of  the  ruler  on  the 
base  line,  and  turn  the  table  about  till  the  sights  are  directed 
to  the  station  B.  Then  placing  the  edge  of  the  ruler  against 
the  needle,  direct  the  sights  in  succession  to  all  the  objects  ob- 
served from  the  other  station,  drawing  lines  from  the  point  A 
in  their  several  directions.  The  intersections  of  these  lines 
with  those  drawn  from  the  point  B  will  determine  the  posi- 
tions of  the  several  objects  on  the  map. 

In  this  manner  the  plane  table  may  be  employed  for  filling 
in  the  details  of  a  map  ;  setting  it  up  at  the  most  remarkable 
spots,  and  sketching  by  the  e,ye  what  is  not  necessary  should 
be  more  particularly  determined,  the  paper  will  gradually  be- 
come a  representation  of  the  country  to  be  surveyed. 

To  determine  the  Depth  of  Water. 

(186.)  Let  signals  be  established  on  the  principal  shoals  anc! 
along  the  edges  of  the  channel,  by  erecting  poles  or  anchoring 
buoys,  and  let  their  bearings  be  observed  from  two  stations  of 
the  survey.  Then  in  each  triangle  there  will  be  known  one 
side  and  the  angles,  from  which  the  other  sides  may  be  com- 
puted, and  their  positions  thus  become  known.  Then  ascer 


U34  TRIGONOMETRY. 

tain  the  precise  depth  of  water  at  each  of  the  buoys,  and  pro- 
ceed in  this  manner  to  determine  as  many  points  as  may  he 
thought  necessary. 

If  an  observer  is  stationed  with  a  theodolite  at  each  extremi- 
ty of  the  base  line,  we  may  dispense  with  the  erection  of  per- 
manent marks  upon  the  water.  One  observer  in  a  boat  may 
make  a  sounding  for  the  depth  of  water,  giving  a  signal  at  the 
same  instant  to  two  observers  at  the  extremities  of  the  base 
line.  The  direction  of  the  boat  being  observed  at  that  instant 
from  two  stations,  the  precise  place  of  the  boat  can  be  com- 
puted. In  this  way  soundings  may  be  made  with  great  ex- 
pedition. 

There  is  also  another  method,  still  more  expeditious,  which 
may  afford  results  sufficiently  precise  in  some  cases.  Let  a 
boat  be  rowed  uniformly  across  the  harbor  from  one  station  tc 
another,  for  example,  from  D  to  Gr  (see  fig.  on  p.  131),  and  let 
a  series  of  soundings  be  made  as  rapidly  as  possible,  and  the 
instant  of  each  sounding  be  recorded.  Then,  knowing  the  en- 
tire length  of  the  line  DG-,  and  the  time  of  rowing  over  it,  we 
may  find  by  proportion  the  approximate  position  of  the  boat  at 
each  sounding. 

If  the  soundings  are  made  in  tide  waters,  the  times  of  high 
water  should  be  observed,  and  the  time  of  each  sounding  be 
recorded,  so  that  the  depth  of  water  at  high  or  low  tide  may 
be  computed.  In  the  maps  of  the  United  States  Coast  Survey, 
the  soundings  are  all  reduced  to  low-water  mark,  and  the  num- 
ber of  feet  which  the  tide  rises  or  falls  is  noted  upon  the  map 

(187.)  The  results  of  the  soundings  may  be  delineated  upon 
a  map  in  the  same  manner  as  the  observations  of  level  on  page 
124.  We  draw  lines  joining  all  those  points  where  the  depth 
of  water  is  the  same,  for  example,  20  feet.  Such  a  line  is  s-een 
to  be  an  undulating  line  running  in  the  direction  from  E  to  Gr. 
We  draw  another  line  connecting  all  those  points  where  the 
depth  of  water  is  40  feet.  This  line  runs  somewhat  to  the 
east  of  the  former  line,  but  nearly  parallel  with  it.  We  draw 
other  lines  for  depths  of  60  feet,  &c.  The  lines  being  thus 
drawn,  a  mere  glance  at  the  map  will  show  nearly  the  depth 
of  water  at  any  point  of  the  harbor. 


BOOK  V. 

N  A  V I  G  A  T  I  0  N. 

NAVIGATION  is  the  art  of  conducting  a  ship  at  soa 
from  <  ne  port  to  another. 

Tli^re  are  two  methods  of  determining  the  situation  of  a 
vessel  at  sea.  The  one  consists  in  finding  by  astronomical  ob- 
servations her  latitude  and  longitude ;  the  other  consists  in 
measuring  the  ship's  course,  and  her  progress  every  day  from 
the  time  of  her  leaving  port,  from  which  her  place  may  be 
computed  by  trigonometry.  The  latter  method  is  the  one  to 
be  now  considered. 

(189.)  The  figure  of  the  earth  is  nearly  that  of  a  sphere,  and 
in  navigation  it  is  considered  perfectly  spherical.  The  earth's 
axis  is  the  diameter  around  which  it  revolves  once  a  day. 
The  extremities  of  this  axis  are  the  terrestrial  poles  ;  one  is 
called  the  north  pole,  and  the  other  the  south  pole. 

The  equator  is  a  great  circle  perpendicular  to  the  earth's 
axis. 

Meridians  are  great  circles  passing  through  the  poles  of  the 
earth.  Every  place  on  the  earth's  surface  has  its  own  meridian. 

(190.)  The  longitude  of  any  place  is  the  arc  of  the  equator 
intercepted  between  the  meridian  of  that  place  and  some  as- 
sumed meridian  to  which  all  others  are  referred.  In  most 
countries  of  Europe,  that  has  been  taken  as  the  standard  me- 
ridian which  passes  through  their  principal  observatory.  The 
English  reckon  longitude  from  the  Observatory  of  Greenwich ; 
and  in  the  United  States,  we  have  usually  adhered  to  the  En- 
glish custom,  though  we  believe  the  time  has  come  when  longi- 
tude should  be  reckoned  from  the  Observatory  of  Washington. 

Longitude  is  usually  reckoned  east  and  west  of  the  first  me- 
ridian, from  0°  to  180°. 

The  difference  of  longitude  of  two  places  is  the  arc  of  the 
equator  included  between  their  meridians.  It  is  equal  to  the 


136  TRIGONOMETRY 

difference  c  f  their  longitudes  if  they  are  on  the  same  side  of 
the  first  meridian,  and  to  the  sum  of  their  longitudes  if  on  op. 
posite  sides. 

(191.)  The  latitude  of  a  place  is  the  arc  of  the  meridian  pass- 
ing  through  the  place,  which  is  comprehended  between  that 
place  and  the  equator. 

Latitude  is  reckoned  north  and  south  of  the  equator,  from 
0°  to  90°. 

Parallels  of  latitude  are  the  circumferences  of  small  circles 
parallel  to  the  equator. 

The  difference  of  latitude  of  two  places  is  the  arc  of  a  me- 
ridian included  between  the  parallels  of  latitude  passing 
through  those  places.  It  is  equal  to  the  difference  cf  their 
latitudes  if  they  are  on  the  same  side  of  the  equator,  and  to 
the  sum  of  their  latitudes  if  on  opposite  sides. 

The  distance  is  the  length  of  the  line  which  a  vessel  de- 
scribes in  a  given  time. 

The  departure  of  two  places  is  the  distance  of  either  place 
from  the  meridian  of  the  other.  If  the  two  places  are  01; 
the  same  parallel,  the  departure  is  the  distance  between  the 
places.  Otherwise,  we  divide  the  distance  AB  into  portion? 
A£,  be,  cd,  &c.,  so  small 
that  the  curvature  of  the 
earth  may  be  neglected. 
Through  these  points 
we  draw  the  meridians 
P£,  PC,  &c.,  and  the  par- 
allels be,  cf,  &c.  Then 
the  departure  for  A.b  is 
eh,  for  be  it  is  fc  ;  and 
the  whole  departure  from  A  to  B  is  eb+fc+gd+hB ;  that  is, 
the  sum  of  the  departures  corresponding  to  the  small  portions 
into  which  the  distance  is  divided. 

Distance,  departure,  and  difference  of  latitude  are  measured 
in  nautical  miles,  one  of  which  is  the  60th  part  of  a  degree  at 
the  equator.  A  nautical  mile  is  nearly  one  sixth  greater  than 
an  English  statute  mile. 

The  course  of  a  ship  is  the  angle  which  the  ship's  path  makes 
with  the  meridian.  A  ship  is  said  to  continue  on  the  same 


NAVIGATION. 


13-5 


course  when  she  cuts  every  meridian  which  she  crosses  at  the 
same  angle.  The  path  thus  described  is  not  a  straight  line, 
but  a  curve  called  a  rhumb-line. 

The  course  of  a  ship  is  given  by  the  mariner's  compass. 

(192.)  The  mariner's  compass  consists  of  a  circular  piece  of 
paper,  called  a  card,  attached  to  a  magnetic  needle,  which  ia 
balanced  on  a  pin  so  as  to  move  freely  in  any  direction.  Di 
rectly  over  the  needle,  a  line  is  drawn  on  the  card,  one  end  ol 
which  is  marked  N,  and  the  other  S.  The  circumference  is 
divided  into  thirty-two  equal  parts  called  rhumbs  or  points, 
each  point  being  subdivided  into  four  equal  parts  called  quarter 
points. 

The  points  of  the  compass  are  designated  as  follows,  begin- 
ning at  north  and  go- 
ing east:  north,  north 
by  east,  north-north- 
east, northeast  by 
north,  northeast,  and 
so  on,  as  shown  in  the 
annexed  figure. 

The  interval  be- 
tween two  adjacent 
points  is  11°  15', 
which  is  the  eighth 
part  of  a  quadrant. 
On  the  inside  of  the 
compass-box  a  black 
line  is  drawn  perpen- 
dicular to  the  horizon,  and  the  compass  should  be  so  placeo 
that  a  line  drawn  from  this  mark  through  the  center  of  the 
card  may  be  parallel  to  the  keel  of  the  ship.  The  part  of  the 
card  which  coincides  with  this  mark  will  then  show  the  point 
of  the  compass  to  which  the  keel  is  directed.  The  compass  is 
suspended  in  its  box  in  such  a  manner  as  to  maintain  a  hori- 
zontal position,  notwithstanding  the  motion  of  the  ship. 

The  following  taUe  shows  the  number  of  degrees  and  min- 
utes corresponding  to  each  point  of  the  compass  ; 


L3S 


TRIGONOMETRY 


North. 

Pts. 

Pts. 

South. 

N.  by  E. 

N.N.E. 

N.byW. 

N.N.W. 

1 

2 

11°  15' 

22°  30' 

1 

2 

S.  by  E. 

S.S.E. 

S.byW. 
S.S.W. 

N.E.byN. 
N.E. 

N.W.byN. 
N.W. 

3 
4 

33°  45' 
45°  0' 

3 
4 

S.E.byS. 
S.E. 

S.W.byS. 

s.w. 

N.E.byE. 
E.N.E. 

N.W.byW. 
W.N.W. 

5 
6 

56°  15' 
67°  30' 

5 

6 

S.E.byE. 
E.S.E. 

S.W.byW. 
W.S.W. 

E.byN. 
East. 

W.byN. 
West. 

7 

8 

78°  45' 
90°  0' 

7 
8 

E.byS. 
East. 

W.byS. 

"West. 

(193.)  The  ship's  rate  of  sailing  is  measured  by  a  log-line, 
The  log-line  is  a  cord  about  300  yards  long,  which  is  wound 


round  a  reel,  one  end  being  attached  to  a  piece  of  thin  board 
called  a  log.  This  board  is  in  the  form  of  a  sector  of  a  circle, 
the  arc  of  which  is  loaded  with  lead  sufficient  to  give  the  board 
a  vertical  position  when  thrown  upon  the  water.  This  is  de- 
signed to  prevent  the  log  from  being  drawn  along  after  the 
vessel  while  the  line  is  running  off  the  reel. 

The  time  is  measured  by  a  sand-glass,  through  which  the 
•sand  passes  in  half  a  minute,  or  the  120th  part  of 
an  hour. 

The  log-line  is  divided  into  equal  parts  called 
knots,  each  of  which  is  50  feet,  or  the  120th  part 
of  a  nautical  mile.  Now,  since  a  knot  has  the 
same  ratio  to  a  nautical  mile  that  half  a  minute 
has  to  an  hour,  it  follows,  that  if  the  motion  of  a 
ship  is  uniform,  she  sails  as  many  miles  in  an  hour 
as  she  does  knots  in  half  a  minute.  If,  then,  seven  knots  are 
observed  to  run  off  in  half  a  minute,  the  ship  is  sailing  at  the 
rate  of  seven  miles  an  hour. 

PLANE  SAILING. 

(194.)  Plane  sailing  is  the  method  of  calculating  a  ship's 
place  at  sea  by  means  of  the  properties  of  a  plane  triangle. 
The  particulars  which  are  given  or  required  are  four,  viz.,  the 


NAVIGATION. 


distance,  course,  difference  of  latitude,  and  departure.     Of 
these,  any  two  being  given,  the  others  may  be  found. 

Let  the  figure  EPQ,  represent  a  portion  of  the  earth's  sur- 
face, P  the  pole,  and  EQ, 
the  equator.  Let  AB 
be  a  rhumb-line,  or  the 
track  described  by  a  ship 
in  sailing  from  A  to  B 
on  a  uniform  course. 
Let  the  whole  distance 
be  divided  into  portions 
A&,  be,  &c.,  so  small 
that  the  curvature  of  the  earth  may  be  neglected.  Through 
the  points  of  division  draw  the  meridians  P#,  PC,  &c.,  and  th<3 
parallels  eb\  fc,  &c.  Then,  since  the  course  is  every  where 
the  same,  each  of  the  angles  ehb,  fbc>  &c.,  is  equal  to  the 
course.  The  distances  Ae,  bf,  &c.,  are  the  differences  of  lati- 
tude of  A  and  &,  b  and  c,  &c.  Also,  eb,  ft,  &o.,  are  the  de- 
partures for  the  same  distances.  Hence  the  difference  of  lati- 
tude from  A  to  B  is  equal  to 

Ke+bf+cg+dh,  c'  ^ 

and  the  departure  is  equal  to 
eb+fc+gd+hB. 

Construct  the  triangle  A'B'C'  so  that  A.'b'e' 
shall  be  equal  to  Pd>e,  b'c'f  shall  be  equal  to 
bcf,  c'd'g'  equal  to  cdg,  and  d'E'/i'  equal  to 
dltfi.     Then  A'B'  represents  the  distance  sail-     A' 
ed,  B'A'C'  the  course,  A'C7  the  difference  of  latitude,  and  B'C 
the  departure ;  that  is,  the  distance,  dif- 
ference of  latitude,  and  departure  are  cor- 
rectly represented  by  the  hypothenuse  and 
sides  of  a  right-angled  triangle,  of  which 
the  angle  opposite  to  the  departure  is  the 
course.     Of  these  four  quantities,  any  two 
being  given,  the  others  may  be  found. 

Plane  sailing  does  not  assume  the  earth's 
surface  to  be  a  plane,  and  does  not  involve 
any  error  even  in  great  distances. 


C      Dcpartttre 


140  TRIGONOMETRY. 

EXAMPLES. 

1.  A  ship  sails  from  Yera  Cruz  N.E.  by  N".  74  miles.     R<y 
quired  her  departure  and  difference  of  latitude 

According  to  the  principles  of  right-angled  triangles,  Art.  44. 
Radius  :  distance  :  :  sin.  course  :  departure. 

:  :  cos.  course  :  diff.  latituat, 
The  course  is  three  points,  or  33°  45' ;  hence  we  obtain 

Departure      =41.11  miles. 
Diff.  latitude = 61.53  miles. 

2.  A  ship  sails  from  Sandy  Hook,  latitude  40°  28'  N.,  upon 
a  course  B.S.B.,  till  she  makes  a  departure  of  500  miles.   "What 
distance  has  she  sailed,  and  at  what  latitude  has  she  arrived  ? 

"By  Trigonometry,  Art.  44, 

Sin.  course  :  departure  :  :  radius  :  distance, 

: :  cos.  course  ;  diff.  latitude 
Ans.  Distance        =541.20  miles. 

Diff.  latitude=207.11  miles,  or  3°  2?'. 
Hence  the  latitude  at  which  she  has  arrived  is  37°  1'  N. 

3.  The  bearing  of  Sandy  Hook  from  Bermuda  is  N.  42°  56' 
W.,  and  the  difference  of  latitude  486  miles.     Required  the 
distance  and  departure. 

By  Trigonometry,  Art.  46, 

Radius  :  diff.  latitude  :  :  tang,  course  :  departure, 
:  :  sec.  course  :  distance. 
Ans.  Distance   =663.8  miles. 
Departure =452.1  miles. 

4.  A  ship  sails  from  Bermuda,  latitude  32°  22'  N.,  a  distance 
of  666  miles,  upon  a  course  between  north  and  east,  until  she 
finds  her  departure  444  miles.     "What  course  has  she  sailed, 
and  what  is  her  latitude  ? 

By  Trigonometry,  Art.  44, 

Distance  :  radius  :  :  departure  :  sin.  course, 
Radiiis  :  distance  :  :  cos.  course  :  diff.  latitude. 

Ans.  Latitude=      40°  38'  N. 
Course    =N.  41°  49' E 

5.  The  distance  from  Yera  Cruz,  latitude  19°  12'  N.,  to  Pen- 
bacola,  latitude  30°  19'  N.,  is  820  miles.     Required  the  bear- 
ing  and  departure. 


NAVIGATION.  Ill 

By  Trigonometry,  Art.  45, 

Distance  :  radius  : :  diff.  latitude  :  cos.  course, 
Radius  :  distance  :  :  sin.  course  :  departure. 

Ans.  Bearing    =N.  35°  34'  E. 
Departure =476. 95  miles 

(5.  A  ship  sails  from  Sandy  Hook  upon  a  course  between 
and  east  to  the  parallel  of  35°,  when  her  departure  waa 
300  miles.     Required  her  course  and  distance. 
By  Trigonometry,  Art.  47, 

Diff.  latitude  :  radius  : :  departure  :  tang',  course, 
Radius  :  diff.  latitude  :  :  sec.  course  :  distance. 

Ans.  Course  S.  42°  27'  E. 
Distance  444.5  miles 

TRAVERSE  SAILING. 

(195.)  A  traverse  is  the  irregular  path  of  a  ship  when  sail- 
ing on  different  courses. 

.  The  object  of  traverse  sailing-  is  to  reduce  a  traverse  to  a 
single  course,  when  the  distances  sailed  are  so  small  that  the 
curvature  of  the  earth  may  be  neglected.  When  a  ship  sails 
on  different  courses,  the  difference  of  latitude  is  equal  to  the 
difference  between  the  sum  of  the  northings  and  the  sum  of  the 
southings ;  and,  neglecting  the  earth's  curvature,  the  departure 
is  equal  to  the  difference  between  the  sum  of  the  eastings  and 
the  sum  of  the  westings.  If,  then,  the  difference  of  latitude  and 
the  departure  for  each  course  be  taken  from  the  traverse  table, 
and  arranged  in  appropriate  columns,  the  difference  of  latitude 
for  the  whole  time  may  be  obtained  exactly,  and  the  departure 
nearly,  by  addition  and  subtraction ;  and  the  corresponding 
distance  and  course  may  be  determined  as  in  plane  sailing. 

EXAMPLES. 
1.  A  ship  sails  on  the  following  successive  tracks ; 

1.  N.B.      23  miles. 

2.  E.S.E.    45      " 

3.  E.  by  N.  34      « 

4.  North      29      « 

5.  N.  by  W.  31      " 

6.  N.N.E.    17      « 

Find  the  course  and  distance  for  the  whole  traverse. 


142 


TRIGONOMETRY. 


We  form  a  table  as  below,  entering  the  courses  from  th« 
table  of  rhumbs,  page  138,  and  then  enter  the  latitudes  and 
departures  taken  from  the  traverse  table. 

Traverse  Table. 


No.  |                Course. 

Distance. 

N. 

s. 

E. 

w. 

1 

N.  45°  E. 

23 

16.26 

16.26 

2 

S.  67°  30'  E. 

45 

17.22 

41.57 

3 

N.  78°  45'  E. 

34 

6.63 

33.35 

4 

North. 

29 

29.00 

5 

N.  11°  15'  W. 

31 

30.40 

6.05 

6 

N.  22°  30'  E. 

17 

15.71 

6.51 

Sum  of  columns 


98.00      17.22 
17.22 


97.69      6.05 
6.05 


Diff.  latitude      .     .     .     . -80.78  N.    Dep.-91.64E. 

Hence  the  course  is  found  by  plane  sailing  N.  48°  36'  E., 
and  the  distance  =122.2  miles. 

The  proportions  are 

Diff.  latitude  :  radius  :  :  departure  :  tang,  course, 
Radius  :  diff.  latitude  : :  sec.  course  :  distance. 

2.  A  ship  leaving  Sandy  Hook  makes  the  following  courses 
and  distances : 

1.  S.E.      25  miles. 

2.  E.S.E.    32      " 

3.  East     17      " 

4.  E.byS.  51      « 

5.  South    45      " 

6.  S.byE.  63      » 

Required  her  latitude,  the   distance   made,   and  the  direct 
Bourse. 

Ans.  Latitude=38°  1'  N. 
Distance =193.7  miles. 
Course    =S.  40°  47'  E. 

3.  A  ship  from  Pensacola,  latitude  30°  19',  sails  on  1he  fol- 
lowing successive  cours3S : 

1        South       48  miles. 

2.  S.S.W.      23      " 

3.  S.W.        32      " 


NAVIGATION. 


143 


66  miles. 
14      " 
45      " 
21      « 
32      « 


4    S.W.  by  S.  76  miies. 

5.  West       17      " 

6.  W.S.W.     54      « 
Required  her  latitude,  direct  course,  and  distance. 

-4«5.  Latitude-     27°  23'  N. 
Course    =  S.3S°39'W. 
Distance  =225.0  miles. 

4.  A  ship  from  Bermuda,  latitude  32°  22',  sails  on  the  fol- 
lowing successive  courses : 

1.  N.E. 

2.  N.N.B. 

3.  N.E.  by  E, 

4.  East 

5.  E.byN. 

Required  her  latitude,  direct  course,  and  distance. 

Ans.  Latitude-  33°  53'  N 
Course  =N.  57°  22'  E 
Distance =168.4  miles. 

(196.)  When  the  water  through  which  a  ship  is  moving  ha& 
a  progressive  motion,  the  ship's  progress  is  affected  in  the  same 
manner  as  if  she  had  sailed  in  still  water,  with  an  additional 
course  and  distance  equal  to  the  direction  and  motion  of  the 
current. 

Ex.  5.  If  a  ship  sail  125  miles  N.N.E.  in  a  current  which 
sets  W.  by  N.  32  miles  in  the  same  time,  required  her  true 
course  and  distance. 

Form  a  traverse  table  containing  the  course  sailed  by  the 
ship  and  the  progress  of  the  current,  and  find  the  difference 
of  latitude  and  departure.  The  resulting  course  and  distance 
is  found  as  in  the  preceding  examples. 

Traverse  Table. 


Courses. 

Distance. 

N. 

E. 

w. 

N.  22°  30'  E 
N.  78°  45'  W. 

125 
32 

115.49 
6.24 

47.84 

31.39 

Diff.  latitude 


Departure 


=  121.73     47.84     31.39 

31.39 
r-:  16.45  E. 


144  TRIGONOMETRY. 

Hence  the  course  is  found  by  plane  sailing  N.  7°  42'  E,,  and 
the  distance^  122.8  miles. 

Ex.  6.  A  ship  sails  S.  by  E.  for  two  hours  at  the  rate  of  i) 
miles  an  hour ;  then  S.  by  "W.  for  five  hours  at  the  rate  of  8 
miles  an  hour ;  and  during  the  whole  time  a  current  sets  "W. 
by  N.  at  the  rate  of  two  and  a  half  miles  an  hour,  Required 
the  direct  course  and  distance. 

Ans.  The  course  is  S.  21°  51'  W. 
Distance  57.6  miles. 

PARALLEL  SAILING. 

(197.)  Parallel  sailing"  is  when  a  ship  sails  exactly  east  01 
west,  and  therefore  remains  constantly  on  the  same  parallel 
of  latitude.  In  this  case  the  departure  is  equal  to  the  distance 
sailed,  and  the  difference  of  longitude  may  be  found  by  the  fol- 
lowing 

THEOREM. 

The  cosine  of  the  latitude  of  the  parallel  is  to  radius,  as 
the  distance  run  is  to  the  difference  of  longitude. 

Let  P  be  the  pole  of  the  earth,  C  the  center,  AB  a  portion 
of  the  equator,  and  DE  any  parallel  of  lati-      p 
tude ;    then  will  CA  be  the  radius  of  the 
equator,  and  FD  the  radius  of  the  parallel. 
Let  DE  be  the  distance  sailed  by  the  ship 
on  the  parallel  of  latitude,  then  the  difference 
of  longitude  will  be  measured  by  AB,  the 
arc  intercepted  on  the  equator  by  the  merid- 
ians passing  through  D  and  E. 

Since  AB  and  DE  correspond  to  the  equal  angles  ACB, 
DFE,  they  are  similar  arcs,  and  are  to  each  other  as  their 
radii.  Hence 

FD  :  CA  :  :  arc  DE  :  arc  AB. 

But  FD  is  the  sine  of  PD,  or  the  cosine  of  AD,  that  is,  the 
cosine  of  the  latitude,  and  CA  is  the  radius  of  the  sphere ; 
hence 

Cosine  of  latitude  :  R  :  :  distance  :  diff.  longitude,. 

Cor.  Like  portions  of  different  parallels  of  latitude  are  tr 
each  other  as  the  cosines  of  the  latitudes. 


NAVIGATION. 


The  length  of  a  degree  of  longitude  in  different  parallels  may 
oe  computed  by  thL  theorem.  A  degree  of  longitude  at  the 
equator  being  60  nautical  miles,  a  degree  in  latitude  40°  may 
be  found  by  the  proportion 

R  :  cosine  40°  :  :  60  :  45.96,  the  required  length. 

The  following  table  is  computed  in  the  same  manner. 

(198.)  Table  showing  the  length  of  a  degree  of  longitudt 
for  each  degree  of  latitude. 


Lat. 

Miles. 

Lat 

Miles. 

Lat. 

Miles. 

Lat. 

Miles. 

Lat.j    Miles. 

Lat. 

Miles. 

1|  59.99 

16 

57.68 

31 

51.43 

46 

41.68 

61(29.09 

76' 

14.52! 

2 

59.96 

171  57.38 

32 

50.88 

47 

40.92 

62 

28.17 

77 

13.50 

3 

59.92 

18157.06 

33 

50.32 

48 

40.15 

63 

27.24 

78 

12.47 

4 

59.85 

19 

56.73 

34 

49.74 

49 

39.36 

64 

26.30 

79 

11.45 

5 

59.77 

20 

56.38 

35 

49.15 

50 

38.57 

65 

25.36 

80 

10.42 

6 

59.67 

21 

56.01 

36 

48.54 

51 

37.76 

66 

24.40 

81 

9.39 

7 

59.55 

22 

55.63 

37 

47.92 

52 

36.94 

67 

23.44 

82 

8.35 

8 

59.42 

23 

55.23 

38 

47.28 

53 

36.11 

68 

22.48 

83 

7.31 

9 

59.26 

24 

54.81 

39 

46.63 

54 

35.27 

69 

21.50 

84 

6.27 

10 

59.09 

25|  54.38 

40 

45.96 

55 

34.41 

70 

20.52 

85 

5.23 

11 

58.90 

26 

53.93 

41 

45.28 

56 

33.55 

71 

19.53 

86 

4.19 

12 

58.69 

27 

53.46 

42 

44.59 

57 

32.68 

72 

18.54 

87 

3.14 

13 

58.46 

28 

52.98 

43 

43.88 

58 

31.80 

73 

17.54 

88 

2.09 

14 

58.22 

29 

52.48 

44 

43.16 

59 

30.90 

74 

16.54 

89 

1.05 

15 

57.96 

30 

51.96 

45 

42.43 

60 

30.00 

75 

15.53 

90 

0.00 

Let  ABC  represent  a  right-angled  triangle ;  then,  by  Trig, 
onometry,  Art.  41, 

cos.  B  :  R  :  :  AB  :  BC. 
But,  by  the  preceding  Theorem,  we  have 

cos.  lat.  :  R  :  :  depart.  :  diff.  long-., 
trom  which  we  see  that  if  one  leg  of  a 
right-angled  triangle  represent  the  dis- 
tance run  on  any  parallel,  and  the  adjacent  acute  angle  be 
made  equal  to  the  degrees  of  latitude  of  that  parallel,  then  the 
hypothenuse  will  represent  the  difference  of  longitude. 

EXAMPLES. 

1.  A  ship  sails  from  Sandy  Hook,  latitude  40°  28'  N.,  longi- 
tude 74°  V  AY.,  618  miles  due  east.  Required  her  present 
longitude. 

Cos.  40°  28'  :  R  :  :  618  :  812'.3=13°  32',  the  difference  of 
longitude. 

K 


146  TRIGONOMETRY. 

This,  subtracted  from  74°  1',  leaves  60°  29'  W.,  the  longi- 
tude  required. 

2.  A  ship  in  latitude  40°  saLs  due  east  through  nine  degrees 
of  longitude.     Required  the  distance  run. 

Ans.  413.66  miles. 

3.  A  ship  having  sailed  on  a  parallel  of  latitude  261  miles, 
finds  her  difference  of  longitude  6°  15',     "What  is  her  latitude  ? 

Ans.  Latitude  45°  54'. 

4.  Two  ships  in  latitude  52°  N.,  distant  from  each  other  95 
miles,  sail  directly  south  until  their  distance  is  150  miles 
What  latitude  do  they  arrive  at? 

Ans.  Latitude  13°  34' 

MIDDLE  LATITUDE  SAILING.  . 

(199.)  By  the  method  just  explained  may  be  found  the  lon- 
gitude which  a  ship  makes  while  sailing  on  a  parallel  of  lati- 
tude. "When  the  course  is  oblique,  the  departure  may  be  found 
by  plane  sailing,  but  a  difficulty  is  found  in  converting  this 
departure  into  difference  of  longitude. 

If  a  ship  sail  from  A  to  B,  the  departure  is  equal  to  eb  +  fc 
+gd+hB,  which  is  less  than  AC,  but 
greater  than  DB.  Navigators  have  as- 
sumed that  the  departure  was  equal  to 
the  distance  between  the  meridians  PA, 
PB,  measured  on  a  parallel  EF,  equidis- 
tant from  A  and  B,  called  the  middle  lati- 
tude. 

The  middle  latitude  is  equal  to  half  the 
sum  of  the  two  extreme  latitudes,  if  both 
are  north  or  both  south ;  but  to  half  their  difference,  if  one  is 
north  and  the  other  south. 

The  principle  assumed  in  middle  latitude  sailing  is  not  per 
fectly  correct.     For  long  distances  the  error  is  considerable, 
but  the  method  is  rendered  perfectly  accurate  by  applying  to 
the  middle  latitude  a  correction  which  is  given  in  the  accom 
panying  tables,  page  149. 

(200.)  It  has  been  shown  that  when  a  ship  sails  upon  an 
oblique  course,  the  distance,  departure,  and  difference  of  lati 
tude  may  be  represented  by  the  sides  of  a  right-angled  trian 


NAVIGATION. 


14? 


gfo.  The  difference  of  longitude  is  derived  from  the  departure/ 
in  the  same  manner  as  in  parallel  sailing,  the  ship  being  sup- 
posed to  sail  on  the  middle  latitude  parallel.  Hence,  if  we 
combine  the  triangle  ABC  for  plane  sailing 
with  the  triangle  BCD  for  parallel  sailing, 
we  shall  obtain  a  triangle  ABD,  by  which 
all  the  cases  of  middle  latitude  sailing  may 
be  solved. 

In  the  triangle  BCD, 

Cos.  CBD  :  BC  :  :  R  :  BD  ; 
that  is,  cosine  of  middle  latitude  is  to  the 
departure,  as  radius  is  to  the  difference  of 
longitude. 

In  the  triangle  ABD,  since  the  angle  D  is 
the  complement  of  CBD,  which  represents  the  middle  latitude, 
we  have 

Sin.  D  :  AB  :  :  sin.  A  :  BD  ; 

that  is,  cosine  of  middle  latitude  is  to  the  distance,  as  the  sine 
of  the  course  is  to  the  difference  of  longitude. 

In  the  triangle  ABC,  we  have  the  proportion 
AC  :  BC  :  :  R  :  tang.  A. 

But  we  have  before  had  the  proportion 

Cos.  CBD  :  BC  :  :  R  :  BD. 

The  means  being  the  same  in  these  two  proportions,  we  have 

Cos.  CBD  :  AC  :  :  tang.  A  :  BD  ; 

that  is,  cosine  of  middle  latitude  is  to  the  difference  of  lati- 
tude, as  the  tangent  of  the  course  is  to  the  difference  of  lon- 
gitude. 

The  middle  latitude  should  always  be  corrected  according 
to  the  table  on  page  149.  The  given  middle  latitude  is  to  be 
looked  for  either  in  the  first  or  last  vertical  column,  opposite 
to  which,  and  under  the  given  difference  of  latitude,  is  inserted 
the  proper  correction  in  minutes,  which  must  be  added  to  the 
middle  latitude  to  obtain  the  latitude  in  which  the  meridian 
listance  is  exactly  equal  to  the  departure.  Thus,  if  the  mid- 
le  latitude  is  41°,  and  the  difference  of  latitude  14°,  the  cor- 
rection will  be  found  to  be  25',  which,  added  to  the  middle 
latitude,  gives  the  corrected  middle  latitude  41°  25'. 


148  TRIGONOMETRY. 

EXAMPLES. 

1.  Find  tho  bearing  and  distance  of  Liverpool,  latitude  53^ 
22'  N.,  longitude  2°  52'  "W.,  from  New  York,  latitude  40°  42'  NM 
longitude  74°  V  W. 

Here  are   given  two  latitudes  and  longitudes  to  find  thu 
course  and  distance. 

The  difference  of  latitude  is      ....  12°  40'=  760'. 

The  difference  of  longitude  is   .     .     .     .  71°    9'=4269'. 

The  middle  latitude  is 47°    2'. 

To  which  add  the  correction  from  p.  149          22'. 

The  corrected  middle  latitude  is    ...  47°  24'. 

Then,  according  to  the  third  of  the  preceding  theorems, 
Diff.  lot. :  cos.  mid.  lat. : :  diff.  long. :  tang-,  court e="N.  75°  16'  E 

To  find  the  distance  by  plane  sailing, 

Cos.  course  :  diff.  latitude  :  :  R  :  distance —2988 .4  miles. 

2.  A  ship  sailed  from  Bermuda,  latitude  32°  22'  N.,  longi- 
tude 64°  38'  W.,  a  distance  of  500  miles,  upon  a  course  "W.N  W 
Required  her  latitude  and  longitude  at  that  timo. 

By  plane  sailing, 

E  :  distance  :  :  cos.  course  :  diff.  latitude^  191.3. 
Therefore  the  required  latitude  is  .     .     .     .     .  35°  33'  ; 

the  middle  latitude 33°  58' ; 

and  the  corrected  middle  latitude 33°  59 

Then  we  have 

Cos.  mid.  lat.  :  distance  :  :  sin.  course  :  diff.  long'.=557'.l. 
Therefore  the  longitude  required  is  73°  55'. 

3.  A  ship  sails  southeasterly  from  Sandy  Hook,  latitude 
40°  28'  N.,  longitude  74°  V  W.,  a  distance  of  395  miles,  when 
her  latitude  is  34°  40'  N.     Required  her  course  and  longitude 

Ans.  Course    S.  28°  14'  E. 
Longitude  70°    5'  "W. 

4.  A  ship  sails  from  Brest,  latitude  48°  23'  N.,  longitude 
4°  29'  W.,  upon  a  course  W.S.W.,  till  her  departure  is  556 
miles.     Required  the  distance  sailed  and  the  place  of  the  ship, 

Ans.  Distance  601.8  miles. 
Latitude  44°  33'  N. 
Longitude  17°  57' W. 


^NAVIGATION. 


14Si 


MERCATOR'S  SAILING 


(201.)  Mercator's  sailing  is  a  method  of  computing  differ- 
ence of  iDngitude  on  the  principles  of  Mercator's  chart.  On 
this  chart,  the  meridians,  instead  of  converging  toward  the 
poles  as  they  do  on  the  globe,  are  drawn  parallel  to  each  other, 
by  wnich  means  the  distance  of  the  meridians  is  every  when* 


00°       320° 


made  too  great  except  at  the  equator.  To  compensate  for 
this,  in  order  that  the  outline  of  countries  may  not  be  too 
much  distorted,  the  degrees  of  latitude  are  proportionally  en- 
larged, so  that  the  distance  between  the  parallels  of  latitude 
increases  from  the  equator  to  the  poles.  In  latitude  60°  the 
distance  of  the  meridians  is  twice  as  great,  compared  with  a 
degree  at  the  equator,  as  it  is  upon  a  globe,  and  a  degree  of 
latitude  is  here  represented  twice  as  great  as  near  the  equator 
The  diameter  of  an  island  in  latitude  60°  is  represented  twice 
as  great  as  if  it  was  on  the  equator,  and  its  area  four  times 
too  great.  In  latitude  70°  32'  the  distance  of  the  meridians 
is  three  times  too  great,  in  latitude  75°  31'  four  times  too  great 
and  so  on,  by  which  means  the  relative  dimensions  of  coun- 
tries in  high  latitudes  is  exceedingly  distorted.  On  this  ac* 
count  it  is  not  common  to  extend  the  chart  beyond  latitude  75°. 
(202.)  The  distance  of  any  parallel  upon  Mercator's  chart 
from  the  equator  has  been  computed,  and  is  exhibited  in  the 


150  TRIGONOMETRY. 

accompanying  tables,  pages  142-8,  which  is  called  a  Table  of 
Meridional  Parts.  This  table  may  be  computed  in  the  fol« 
lowing  manner  : 

According  to  Art.  197,  cosine  of  latitude  is  to  radius,  as  the 
departure  is  to  the  difference  of  longitude ;  that  is,  as  a  part 
of  a  parallel  of  latitude  is  to  a  like  part  of  the  equator,  of  any 
meridian. 

But  by  Art.  28,  cosine  :  R  :  :  R  :  secant ;  hence 

1'  of  a  parallel  :  V  of  a  meridian  :  :  R  :  sec.  latitude. 
But  on  Mercator's  chart  the  distance  between  the  meridians 
is  the  same  in  all  latitudes  ;  that  is,  a  minute  on  a  parallel  of 
latitude  is  equal  to  a  minute  at  the  equator,  or  a  geographical 
mile.     Hence  the  length  of  one  minute,  on  any  part  of  a  me- 
ridian, is  equal  to  the  secant  of  the  latitude.     Thus, 
The  first  minute  of  the  meridian  =  the  secant  of  1' ; 
second         "  "         =  "  2', 

third  "  "         =  "  3', 

&c.,  &c. 

The  table  of  meridional  parts  is  formed  by  adding  together 
the  minutes  thus  found.     Thus, 
Mer.  parts  of  1'— sec.  1' ; 
Mer.  parts  of  2'= sec.  I'+sec.  2' ; 
Mer.  parts  of  3'=sec.  I'+sec.  2'+sec.  3'; 
Mer.  parts  of  4'— sec.  I'+sec.  2'+sec.  3'+sec.  4', 

&c.,  &c.,  &c. 

Since  the  secants  of  small  arcs  are  nearly  equal  to  radius? 
or  unity,  if  the  meridional  parts  are  only  given  to  one  tenth 
of  a  mile,  we  shall  have 

The  meridional  parts  of  l'=1.0  mile  ; 
u  u  u        2/:=2  0     u 

"  "  "        3'=3.0     " 

"  "  "         4'=4.0     "     &c., 

as  shown  in  the  table  on  page  142. 

At  2°  33'  the  sum  of  the  small  fractions  omitted  becomes 
greater  than  half  of  one  tenth,  and  the  meridional  parts  ol 
2°  33'  is  153.1 ;  that  is,  the  meridional  parts  exceed  by  one 
tenth  of  a  mile  the  minutes  of  latitude.  At  3°  40'  the  excess 
is  two  tenths  of  a  mile ;  at  4°  21'  the  excess  is  three  tenths ; 


NAVIGATION.  151 

and  as  the  latitude  increases,  the  meridional  pares  increase 
more  rapidly,  as  is  seen  from  the  table. 

An  arc  of  Mercator's  meridian  contained  between  two  par- 
allels of  latitude  is  called  meridional  difference  of  latitude  Tt 
is  faund  by  subtracting  the  meridional  parts  of  the  less  latitude 
from  the  meridional  parts  of  the  greater,  if  both  are  north  or 
south,  or  by  adding  them  together  if  one  is  north  and  the  other 
south.  Thus, 

The  lat.  of  New  York  is  40°  42';  meridional  parts =2677.8, 
<  New  Orleans  29°  57' ;  "  "  1884.9. 

The  true  diff.  of  lat.  is  10°  45' ;  mer.  diff.  lat.  is       792.9 

If  one  latitude  and  the  meridional  difference  of  latitude  be 
given,  the  true  difference  of  latitude  may  be  found  by  reversing 
this  process.  Thus, 

The  meridional  parts  for  New  Orleans      .     .     .    /  =  1884.9. 

Meridional  difference  of  latitude  between  New  ) 
York  and  New  Orleans 

Therefore  the  meridional  parts  for  New  York  =2677.8, 
and  the  corresponding  latitude  from  the  table  is  40°  42'. 

(203.)  If  we  take  the  figure  ABC  for 

i  •!•  l>iffi' 

plane  sailing,  as  on  page  139,  and  pro- 

duce  AC  to  E,  making  AE  equal  to  the 
meridional  difference  of  latitude >  then 
will  DE  represent  the  difference  of  lon- 
gitude corresponding  to  the  departure 
BC.  For  we  have  seen  (Art.  202)  that 
the  departure  is  to  the  difference  of  lon- 
gitude as  radius  is  to  the  secant  of  lati- 
tude, which  is  also  the  ratio  of  the  true 
difference  of  latitude  to  the  meridional  difference  of  latitude. 

Now,  from  the  similarity  of  the  triangles  ABC,  ADE,  wo 
have 

AC  :  AE  : :  BC  :  DE  ; 

that  is,  the  true  difference  of  latitude  is  to  the  meridional  dif* 
ference  of  latitude,  as  the  departure  is  to  the  difference  of 
longitude. 

Also,  in  the  triangle  ADE,  we  have 

R  :  tan.  A  :>  AE  :  DE ; 


152  TRIGONOMETRY. 

that  is,  radius  is  tc  the  tangent  of  the  course,  as  the  vierid* 
ional  difference  of  latitude  is  to  the  difference  of  longitude. 

EXAMPLES. 

1.  Find  the  bearing  and  distance  from  Sandy  Hook,  latitude 
40°  28'  N.,  longitude  74°  1'  W.,  to  Havre,  latitude  49°  29'  N., 
longitude  0°  6'  E. 

The  true  difference  of  latitude  is  9°  1'=  541' ; 

meridional  difference  of  latitude  =767.1 ; 

difference  of  longitude  is  74°  7' =4447. 

Hence,  to  find  the  course  by  the  preceding  proportion, 
Mer.  diff.  lat.  :  diff.  long.  :  :  R  :  tan.  course=~N.  80°  13'  K 
To  find  the  distance  by  plane  sailing, 
Cos.  course  :  true  diff.  lat.  : :  R  :  distance=3183.8  miles*. 

2.  Find  the  bearing  and  distance  from  Nantucket  Shoals,  in 
latitude  41°  4'  N.,  longitude  69°  55'  W.,  to  Cape  Clear,  in  lati- 
tude 51°  26'  N.,  longitude  9°  29'  W. 

Ans.  Course  N.  76°  E. 

Distance  2572.9  miles. 

3.  A  ship  sails  from  Sandy  Hook  a  distance  of  600  miles 
upon  a  course  S.  by  E.     Required  the  place  of  the  ship. 

The  difference  of  latitude  may  be  found  by  plane  sailing, 
the  difference  of  longitude  by  Mercator's  sailing. 

Ans.  Latitude    30°  39'.5  N. 
Longitude  71°  36'.7  W. 

4.  A  ship  sails  from  St.  Augustine,  latitude  29°  52'  N.,  lon- 
gitude 81°  25'  W.,  upon  a  course  N.E.  by  E.,  until  her  lati- 
tude is  found  to  be  34°  40'  N.     "What  is  then  her  longitude, 
and  what  distance  has  she  run  ? 

Ans.  Longitude =72°  55'  W. 
Distance    =518.4  miles. 

5.  A  ship  sails  from  Bermuda  upon  a  course  N.W.  by  W 
until  her  longitude  is  found  to  be  69°  30'  W.     What  is  then 
her  latitude,  and  what  distance  has  she  run  ? 

Ans.  Latitude  35°  4'  N. 
Distance  291.6  miles 

6.  A  ship  sailing  from  Madeira,  latitude  32°  38'  N ,  longi. 
tude  16°  ^5'  W.,  steers  westerly  nntil  her  latitude  is  40°  2'  N., 


ATIGATION. 

and  her  departure  2425  miles.     Required  her  course,  distance 
and  longitude. 

Ans.  Course  N.  79°  37'  W 
Distance  2465.3  miles. 
Longitude  67°  9'.3  W. 

7.  Find  the  bearing  and  distance  from  Sandy  Hook,  latitude 
40°  28'  N.,  longitude  74°  V  W.,  to  the  Cape  of  Good  Hope 
latitude  34°  22'  S.,  longitude  18°  30'  E. 

Ans.  Course 
Distance 


CHARTS. 

(204.)  The  charts  commonly  used  in  navigation  are  plane 
charts,  or  Mercator's  chart.  In  the  construction  of  the  former, 
the  portion  of  the  earth's  surface  which  is  represented  is  sup- 
posed to  he  a  plane.  The  meridians  are  drawn  parallel  to  each 
other,  and  the  lines  of  latitude  at  equal  distances.  The  dis- 
tance between  the  parallels  should  be  to  the  distance  between 
the  meridians,  as  radius  to  the  cosine  of  the  middle  latitude 
of  the  chart.  A  chart  of  moderate  extent  constructed  in  this 
manner  will  be  tolerably  correct.  The  distance  of  the  merid- 
ians in  the  middle  of  the  chart  will  be  exact,  but  on  each  side 
it  will  be  either  too  great  or  too  small. 

When  large  portions  of  the  earth's  surface  are  to  be  repre- 
sented, the  error  of  the  plane  chart  becomes  excessive.  To 
obviate  this  inconvenience  Mercator's  .chart  has  been  con- 
structed. Upon  this  chart  the  meridians  are  represented  by 
parallel  lines,  and  the  distance  between  the  parallels  of  latitude 
is  proportioned  to  the  meridional  difference  of  latitude,  as  rep- 
resented on  page  149. 

We  have  seen  that  the  meridional  difference  of  latitude  is  to 
the  difference  of  longitude  as  radius  is  to  the  tangent  of  the 
course.  Hence,  while  the  course  remains  unchanged,  the  ratio 
of  the  meridional  difference  of  latitude  to  the  difference  of  lon- 
gitude is  constant ;  and,  therefore,  every  rhumb  line  will  be 
represented  on  Mercator's  chart  by  a  straight  line.  This 
property  renders  Mercator's  chart  peculiarly  convenient  ta 
navigators 


1£>4  TRIGONOMETRY. 

The  preceding  sketch  affords  a  very  incomplete  view  of  the 
present  state  of  the  science  of  navigation.  The  most  accurate 
method  of  ascertaining  the  situation  of  a  vessel  at  sea  is  by 
means  of  astronomical  observations.  For  these,  however,  the 
it  must  be  referred  to  some  treatise  on  Astronomy. 


• 


BOOK  VI. 

SPHERICAL  TRIGONOMETRY. 

(205.)  SPHERICAL  trigonometry  teaches  how  to  determine 
the  several  parts  of  a  spherical  triangle  from  having  certain 
parts  given. 

A  spherical  triangle  is  a  portion  of  the  surface  of  a  sphere, 
bounded  "by  three  arcs  of  great  circles,  each  of  which  is  less 
than  a  semicircumference. 

RIGHT-ANGLED  SPHERICAL  TRIANGLES. 
THEOREM  I. 

(206.)  In  any  right-angled  spherical  triangle,  the  sine  of 
the  hypothenuse  is  to  radius,  as  the  sine  of  either  side  is  to 
the  sine  of  the  opposite  angle. 

Let  ABC  be  a  spherical  triangle,  right-angled  at  A ;  then 
will  the  sine  of  the  hypothenuse  BC 
be  to  radius,  as  the  sine  of  the  side 
AC  is  to  the  sine  of  the  angle  ABC. 

Let  D  be  the  center  of  the  sphere ; 
join  AD,  BD,  CD,  and  draw  CE  per- 
pendicular  to  DB,  which  will,  there- 
fore, be  the  sine  of  the  hypothenuse 
BC.  From  the  point  E  draw  the 
straight  line  EF,  in  the  plane  ABD,  perpendicular  to  BD,  and 
join  CF.  Then,  because  DB  is  perpendicular  to  the  two  lines 
CE,  EF,  it  is  perpendicular  to  the  plane  CEF ;  and,  conse- 
quently, the  plane  CEF  is  perpendicular  to  the  plane  ABD 
(Geom.j  Prop.  6,  B.  VII.).  But  the  plane  CAD  is  also  per- 
pendicular to  the  plane  ABD  ;  therefore  their  line  of  common 
section,  CF,  is  perpendicular  to  the  plane  ABD ;  hence  CFD, 
CFE  are  right  angles,  and  CF  is  the  sine  of  the  arc  AC. 

Now,  in  the  right-angled  plane  triangle  CFE, 
CE  :  radius  :  :  CF  :  sine  CEF 


L56  TRIGONOMETRY. 

But  since  CE  and  FE  are  both  at  right  angles  to  DB,  the 
angle  CEF  is  equal  to  the  inclination  of  the  planes  CBD,  ABB ; 
that  is,  to  the  spherical  angle  ABC.     Therefore, 
sine  BC  :  R  : :  sine  AC  :  sine  ABC. 

(207.)  Cor.  1.  In  any  right-angled  spherical  triangle,  tfif 
sines  of  the  sides  are  as  the  sines  of  the  opposite  angles. 

For,  by  the  preceding  theorem, 

sine  BC  :  R  :  :  sine  AC  :  sine  ABC, 
and  sine  BC  :  R  :  :  sine  AB  :  sine  ACB ; 

therefore,  sine  AC  :  sine  AB  :  :  sine  ABC  :  sine  ACB. 

Cor.  2.  In  any  right-angled  spherical  triangle,  the  cosine  of 
either  of  the  sides  is  to  radius,  as  the  cosine  of  the  hypothenuse 
is  to  the  cosine  of  the  other  side. 

Let  ABC  be  a  spherical  triangle,  right-angled  at  A.  De- 
scribe the  circle  DE,  of  which  B  is 
the  pole,  and  let  it  meet  the  three 
sides  of  the  triangle  ABC  produced  in 
I),  E,  and  F.  Then,  because  BD  and 
BE  are  quadrants,  the  arc  DF  is  per- 
pendicular to  BD.  And  since  BAG 

is  a  right  angle,  the  arc  AF  is  per-    D^_^ 

pendicular  to  BD.     Hence  the  point 

F,  where  the  arcs  FD,  FA  intersect 

each  other,  is  the  pole  of  the  arc  BD  (Geom.*  Prop.  5,  Cor.  2, 

B.  IX.),  and  the  arcs  FA,  FD  are  quadrants. 

Now,  in  the  triangle  CEF,  right-angled  at  the  point  E,  ac- 
cording to  the  preceding  theorem,  we  have 

sine  CF  :  R  :  :  sine  CE  :  sine  CFE. 

But  CF  is  the  complement  of  AC,  CE  is  the  complement  of 
BC,  and  the  angle  CFE  is  measured  by  the  arc  AD,  which  is 
the  complement  of  AB.  Therefore,  in  the  triangle  ABC,  wo 
have 

cos.  AC  :  R  : :  cos.  BC  :  cos.  AB. 

Cor.  3.  In  any  right-angled  spherical  triangle,  the  cosine 
of  either  of  the  sides  is  to  radius,  as  the  cosine  of  the  angle 
opposite  to  that  side  is  to  the  sine  of  the  other  angle. 

For,  in  the  triangle  CEF,  we  have 

sine  CF  :  R  :  :  sine  EF  :  sine  ECF. 

But  sine  CF  is  rqual  to  cos.  CA.     EF  is  the  complement  of 


SPHERICAL    TRIGONOMETRY.  157 

ED,  which  measures  the  angle  ABC,  that  is,  sine  EF  is  equal 
to  cos.  ABC,  and  sine  EOF  is  the  same  as  sir.e  ACB ;  there- 
fore, 

cos.  AC  :  R  :  :  cos.  ABC  :  sine  ACB 

THEOREM  II. 

(208.)  In  any  right-angled  spherical  triangle,  the  sine  oj 
either  of  the  sides  about  the  right  angle  is  to  the  cotangent 
of  the  adjacent  angle,  as  the  tangent  of  the  remaining  side 
is  to  radius. 

Let  ABC  be  a  spherical  triangle,  right-angled  at  A ;  then 
will  the  sine  of  the  side  AB  "be  to  the 
cotangent  of  the  angle  ABC,  as  the 
tangent  of  the  side  AC  is  to  radius. 

Let  D  be  the  center  of  the  sphere ; 
join  AD,  BD,  CD  ;  draw  AE  perpen- 
dicular to  BD,  which  will,  therefore, 
be  the  sine  of  the  arc  AB.  Also,  from 
the  point  E  in  the  plane  BDC,  draw 
the  straight  line  EF  perpendicular  to 
BD,  meeting  DC  produced  in  F,  and 
join  AF.  Then  will  AF  be  perpendicular  to  the  plane  ABD 
because,  as  was  shown  in  the  preceding  theorem,  it  is  the  com- 
mon section  of  the  two  planes  ADF,  AEF,  each  perpendiculai 
to  the  plane  ADB.  Therefore  FAD,  FAE  are  right  angles, 
and  AF  is  the  tangent  of  the  arc  AC. 

Now,  in  the  triangle  AEF,  right-angled  at  A,  we  have 
AE  :  radius  :  :  AF  :  tang.  AEF. 

But  AE  is  the  sine  of  the  arc  AB,  AF  is  the  tangent  of  ti.  - 
arc  AC,  and  the  angle  AEF  is  equal  to  the  inclination  of  t3"a 
planes  CBD,  ABD,  or  to  the  spherical  angle  ABC  ;  hence 
sine  AB  :  R  :  :  tang.  AC  :  tang.  ABC. 

And  because,  Art.  28, 

R  :  cot.  ABC  : :  tang.  ABC  :  R ; 
therefore,  sine  AB  :  cot.  ABC  : :  tang.  AC     :  R. 

(209.)  Cor.  1.  In  any  right-angled  spherical  triangle,  '.h* 
cosine  of  the  hypothenuse  is  to  the  cotangent  of  either  of  *ht 
oblique  angles,  as  the  cotangent  of  the  other  oblique  ang  V  z'j 
to  radius 


1 58  TRIGONOMETRY. 

Let  ABC  be  a  spherical  triangle,  right-angled  at  A.     Do- 
scribe  the  circle  DEF,  of  which  B  F 
is  the  pole,  and  construct  the  com- 
plemental  triangle  CEF,  as  in  Cor. 
2,  Theorem  I. 

Then,  in  the  triangle  CEF,  ac- 

jording  to  the  preceding  theorem,  we 

; 

—  ;D 

sine  CE  :  cot.  ECF  ::  tan.  EF  :  R. 

But  CE  is  the  complement  of  BC,  A 

EF  is  the  complement  of  ED,  the  measure  of  the  angle  ABC  ; 
and  the  angle  ECF  is  equal  to  ACB,  being  its  vertical  angle; 
hence 

cos.  BC  :  cot.  ACB  :  :  cot.  ABC  :  R. 

Cor.  2.  In  any  right-angled  spherical  triangle,  the  cosine  of 
either  of  the  oblique  angles  is  to  the  tangent  of  the  adjacent 
side,  as  the  cotangent  of  the  hypothenuse  is  to  radius. 

For,  in  the  complemental  triangle  CEF,  according  to  the 
preceding  theorem,  we  have 

sine  EF  :  cot.  CFE  :  :  tan.  CE  :  R ; 
hence,  in  the  triangle  ABC, 

cos.  ABC  :  tan.  AB  : :  cot.  BC  :  R. 

Napier's  Rule  of  the  Circular  Parts. 

(210.)  The  two  preceding  theorems,  with  their  corollaries, 
are  sufficient  for  the  solution  of  all  cases  of  right-angled  spheri- 
cal triangles,  and  a  rule  was  invented  by  Napier  by  means  of 
which  these  principles  are  easily  retained  in  mind. 

If,  in  a  right-angled  spherical  triangle,  we  set  aside  the  right 
angle,  and  consider  only  the  five  remaining  parts  of  the  trian- 
gle, viz.,  the  three  sides  and  the  two  oblique  angles,  then  the 
two  sides  which  contain  the  right  angle,  and  the  complements 
of  the  other  three,  viz.,  of  the  two  angles  and  the  hypothenuse, 
are  called  the  circular  parts. 

Thus,  in  the  triangle  ABC,  right-angled  at  A,  the  circular 
parts  are  AB,  AC,  with  the  complements  of  B,  BC,  and  C. 

When,  of  the  five  circular  parts,  any  one  is  taken  for  the 

middle  part,  then,  of  the  remaining  four,  the  two  which  are 

'  immediately  adjacent  to  it  on  the  right  and  left  are  called  the 


SPHERICAL    TRIGONOMETRY. 

adjacent  parts  ;  and  the  other  two,  each  of  which  is  separated 
from  the  middle  by  an  adjacent  part, 
are  called  opposite  parts. 

In  every  question  proposed  for  solu- 
tion, three  of  the  circular  parts  are 
concerned,  two  of  which  are  given, 
and  one  required  ;  and  of  these  three,  B 
the  middle  part  must  be  such  that 
the  other  two  may  be  equidistant  from  it;  that  is,  may  be 
either  both  adjacent  or  both  opposite  parts.  The  value  of  the 
part  required  may  then  be  found  by  the  following 

RULE  OF  NAPIER. 

(211.)  The  product  of  the  radius  and  the  sine  of  the  middle 
part,  is  equal  to  the  product  of  the  t&ngents  of  the  adjacent 
parts,  or  to  the  product  of  the  cosines  of  the  opposite  parts. 

It  will  assist  the  learner  in  remembering  this  rule  to  remark, 
that  the  first  syllable  of  each  of  the  words  tangent  and  adja- 
cent contains  the  same  vowel  a,  and  the  first  syllable  of  the 
words  cosine  and  opposite  contains  the  same  vowel  o. 

It  is  obvious  that  the  cosine  of  the  complement  of  an  angle 
is  the  sine  of  that  angle,  and  the  tangent  of  a  complement  is 
a  cotangent,  and  vice  versa. 

In  the  triangle  ABC,  if  we  take  the  side  b  as  the  middle 
part,  then  the  side  c  and  the  complement  of  the  angle  C  are 
the  adjacent  parts,  and  the  complements  of  the  angle  B  and  of 
the  hypothenuse  a  are  the  opposite  parts.  Then,  according  to 
Napier's  rule,  R  sin.  £=tan.  c  cot.  C, 

which  corresponds  with  Theorem  II. 
Also,  by  Napier's  rule, 

R  sin.  b=sin.  a  sin.  B, 
which  corresponds  with  Theorem  I. 

Making  each  of  the  circular  parts  in  succession  the  middle 
part,  we  obtain  the  ten  following  equations  : 

R  sin.  Z>=sin.  a  sin.  B=tan.  c  cot.  C. 
R  sin.  c  =sin.  a  sin.  C=tan.  b  cot.  B. 
R  cos.  B=cos.  b  sin.  C=cot.  a  tan.  c. 
R  cos.  a  =cos.  b  cos.  c  =oot.  B  cot.  C. 
R  cos.  C  — cos.  c  sin.  B=cot.  a  tan.  b. 


160  TRIGONOMETRY. 

(212.)  In  order  to  determine  whether  the  quantity  sought 
is  less  or  greater  than  90°,  the  algebraic  sign  of  each  term 
should  be  preserved  whenever  one  of  them  is  negative.  If  the 
quantity  sought  is  determined  by  means  of  its  cosine,  tangent, 
or  cotangent,  the  algebraic  sign  of  the  result  will  show  whether 
this  quantity  is  less  or  greater  than  90° ;  for  the  cosines,  tan- 
gents, and  cotangents  are  positive  in  the  first  quadrant,  and 
negative  in  the  second.  But  since  the  sines  are  positive  in  both 
the  first  and  second  quadrants,  when  a  quantity  is  determined 
by  means  of  its  sine,  this  rule  will  leave  it  ambiguous  whether 
the  quantity  is  less  or  greater  than  90°.  The  ambiguity  may, 
however,  generally  be  removed  by  the  following  rule. 

In  every  right-angled  spherical  triangle,  an  oblique  angle 
and  its  opposite  side  are  always  of  the  same  species ;  that  is, 
both  are  greater,  or  both  less  than  90°. 

This  follow?  from  the  equation 

R  sin.  b— tan.  c  cot.  C  ; 

where,  sin'"/)  sin.  b  is  always  positive,  tan.  c  must  always  have 
the  same  sign  as  cot.  C ;  that  is,  the  side  c  and  the  opposite 
angle  C  both  belong  to  the  same  quadrant. 

(213.)  "When  the  given  parts  are  a  side  and  its  opposite  an- 
gle, the  problem  admits  of  two  solutions  ;  for  two  right-angled 
spherical  triangles  may  always  be  found,  having  a  side  and  its 
opposite  angle  the  sam.3  in  both,  but  of  which  the  remaining 
sides  and  the  remaining  angle  of  the  one  are  the  supplements 
of  the  remaining  sides  and  the  remaining  angle  of  the  other. 
Thus,  let  BCD,  BAD  be  the  halves  of  two  great  circles,  and 
let  the  arc  CA  be  drawn  perpendicu- 
lar to  BD  ;  then  ABC,  ADC  are  two 
right-angled  triangles,  having  the  side 
AC  common,  and  the  opposite  angle 
B  equal  to  the  angle  D ;  but  the  side  DC  is  the  supplement  of 
BC,  AD  is  the  supplement  of  AB,  and  the  angle  ACD  is  the 
supplement  of  ACB. 

EXAMPLES. 

1.  In  the  right-angled  spherical  triangle  ABC,  there  are 
given  a=63°  56'  and  6=40°.  Required  the  other  side  c,  and 
the  angle?  B  and  C 


SPHERICAL    TRIGONOMETRY.  161 

'lc  find  the  side  c. 

Hoicv  the  circular  parts  concerned  are 
the  two  legs  and  the  complement  of  the 
hypothenase ;  and  it  is  evident  that  if 
the  complement  of  a  he  made  the  mid- 
dle part,  b  and  c  will  he  opposite  parts  ;  B 
hence,  "by  Napier's  rule, 

R  cos.  a=cos.  b  cos.  c  ; 
or,  reducing  this  equation  to  a  proportion, 

'   cos.  b  :  R  :  :  cos.  a  :  cos.  c=54°  59'  49". 

To  find  the  angle  B. 

Here  b  is  the  middle  part,  and  the  complements  of  B  and  a 
are  opposite  parts  ;  hence 

R  sin.  b=cos.  (comp.  a)Xcos.  (comp.  B)=sin.  a  sin.  B, 
or  sin.  a  :  R  :  :  sin.  b  :  sin.  B=45°  41'  25". 

B  is  known  to  he  an  acute  angle,  hecause  its  opposite  side  is 
less  than  90°. 

To  find  the  angle  C. 

Here  the  complement  of  C  is  the  middle  part;  also  b  and 
the  complement  of  a  are  adjacent  parts  ;  hence 

R  cos.  C=cot.  a  tan.  £, 
or  R  :  tan.  b  :  :  cot.  a  :  cos.  0=65°  45'  57". 

Ex.  2.  In  a  right-angled  triangle  ABC,  there  are  given  the 
hypothenuse  a =91°  42',  and  the  angle  B=95°  6'.  Required 
the  remaining  parts. 

To  find  the  angle  C. 

Make  the  complement  of  the  hypothenuse  the  middle  part; 
then  R  cos.  #=cot.  B  cot.  C. 

Whence  C=71°  36'  47". 

To  find  the  side  c. 

Make  the  complement  of  the  angle  B  the  middle  part ;  &rr<J 
we  have  R  cos.  B=cot.  a  tan.  c. 

Whence  c=71°  32'  14'. 

To  find  the  side  b. 

Make  the  side  b  the  middle  part ;  then 

L 


162  TRIGONOMETRY. 

R  sin.  b  —  sin.  a  sin.  B. 
Whence  6=95°  22'  30". 

b  is  known  to  be  greater  than  a  quadrant,  because  its  opposite 
angle  is  obtuse. 

Ex.  3.  In  the  right-angled  triangle  ABC,  the  side  b  is  26° 
4',  and  its  opposite  angle  B  36°.  Required  the  remaining 
parts. 

(a  =-48°  22'  52",  or  131°  37'    8" 

Ans.  ]  c  -42°  19'  17",  or  137°  40'  43". 

(  C  =  64°  14'  26",  or  115°  45'  34 ". 

This  example,  it  will  be  seen,  admits  of  two  solutions,  con- 
formably to  Art.  213. 

Ex.  4.  In  the  right-angled  spherical  triangle  ABC,  there  are 
given  the  side  c,  54°  30',  and  its  adjacent  angle  B,  44°  50'. 
Required  the  remaining  parts. 

(  C=65°  49'  53". 
Ans.  ]a=63°  10'    4". 
(  b  =33°  59'  IV 

Why  is  not  the  result  ambiguous  in  this  case  ? 
Ex.  5.  In  the  right-angled  spherical  triangle  ABC,  the  side 
b  is  55°  28',  and  the  side  c  63°  15'.     Required  the  remaining 
parts. 

(  a  =75°  13'    2". 
Ans.  )  B=58°  25'  47 

(  C  =  67°  27'    1'. 

Ex.  6.  In  the  right-angled  spherical  triangle  ABC,  there  are 
given  the  angle  B=69°  20',  and  the  angle  C=58°  16'.  Re- 
quired  the  remaining  parts. 

/  a=76°  30'  37". 

Ans.  ]£=65°  28'  58". 

(c -55°  47'  46". 

(214.)  A  triangle,  in  which  one  of  the  sides  is  equal  to  a 
quadrant,  may  be  solved  upon  the  same  principles  as  right- 
angled  triangles,  for  its  polar  triangle  will  contain  a  right  «n- 
gle.  See  Geom.,  Prop.  9,  B.  IX. 

Ex.  7.  In  the  spherical  triangle  ABC,  the  side  BC  =  90°,  the 
angle  C=42°  10',  and  the  angle  A=115°  20'.  Required  the 
remaining  parts. 

Taking  the  supplements  of  the  given  parts,  we  shall  have 


SPHERICAL    TRIGONOMETRY  1GH 

in  the  polar  triangle  the  hypothenuse  a/  =  l&03-115°  20'=64° 
40',  and  one  of  the  sides,  c-'  =  180°-420  10'=137°  50',  from 
which,  by  Napier's  rule,  we  find 

B'=115°  23'  20". 
C'=132°  2'  13". 
b'=125°  15'  36". 

Hence,  taking  the  supplements  of  these  arcs,  we  find  the 
parts  of  the  required  triangle  are 

AC -64°  36'  40". 
AB=47°  57'  47". 
B  =54°  44'  24". 

Ex.  8.  In  the  spherical  triangle  ABC,  the  side  AC =90°,  the 
angle  C=69°  13'  46",  and  the  angle  A=72°  12'  4".  Required 
the  remaining  parts. 

(  AB=70°    8'  39". 
Ans.  ]BC=73°  17' 29". 
(  B    =96°  13'  23 

OBLIQUE-ANGLED  SPHERICAL  TRIANGLES. 

THEOREM  III. 

(215.)  In  any  spherical  triangle,  the  sines  of  the  sides  are 
proportional  to  the  sines  of  the  opposite  angles. 

In  the  case  of  right-angled  spherical  triangles,  this  proposi- 
tion has  already  "been  demonstrated. 
Let,  then,  ABC  be  an  oblique-angled 
triangle  ;  we  are  to  prove  that 
sin.  BC  :  sin.  AC  :  :  sin.  A  :  sin.  B. 
Through  the  point  C  draw  an  arc 
of  a  great  circle  CD  perpendicular  to 
A.B.     Then,  in  the  spherical  triangle 
A.CD,  right-angled  at  D,  we  have,  by  Napier's  rule, 

R  sin.  CD— sin.  AC  sin,  A. 
Also,  in  the  triangle  BCD,  we  have 

R  sin.  CD=sin.  BC  sin.  B. 
Hence         sin  AC  sin.  A=sin.  BC  sin.  B, 
or  sin.  BC  :  sin.  AC  : :  sin.  A  :  sin.  B. 

(216.)  Cor.  1.  In  any  spherical  triangle,  the  cosines  of  the 
sides  are  proportional  to  the  cosines  of  the  segments  of  the 
base,  made  by  a  perpendicular  from  the  opposite  angle. 


164  TRIGONOMETRY. 

For,  by  Theorem  I.,  Cor.  2, 

cos.  CD  :  R  :  :  cos.  AC  :  cos.  AD. 
Also,  cos.  CD  :  R  :  :  cos.  BC  :  cos.  BD. 

Hence          cos.  AC  :  cos.  BC  :  :  cos.  AD  :  cos.  BD. 
Cor.  2.   The  cosines  of  the  angles  at  the  base  are  propor 
tional  to  the  sines  of  the  segments  of  the  vertical  angle. 
For,  "by  Theorem  L,  Cor.  3, 

cos.  CD  :  R  :  :  cos.  A  :  sin.  ACD. 
Also,  cos.  CD  :  R  :  :  cos.  B  :  sin.  BCD. 

Hence         cos.  A :  cos.  B  :  :  sin.  ACD  :  sin.  BCD. 

Cor.  3.  The  sines  of  the  segments  of  the  base  are  recipro- 
cally proportional  to  the  tangents  of  the  angles  at  the  base . 

For,  by  Theorem  II, 

sin.  AD  :  R  :  :  tan.  CD  :  tan.  A. 

Also,  sin.  BD  :  R  :  :  tan.  CD  :  tan.  B. 

Hence  sin.  AD  :  sin.  BD  :  :  tan.  B  :  tan.  A. 

Cor.  4.  The  cotangents  of  the  two  sides  are  proportional 
to  the  cosines  of  the  segments  of  the  vertical  angle. 

For,  by  Theorem  II.,  Cor.  2, 

cos.  ACD  :  cot.  AC  :  :  tan.  CD  :  R. 

Also,         cos.  BCD  :  cot.  BC  :  :  tan.  CD  :  R. 

Hence      cos.  ACD  :  cos.  BCD  : :  cot.  AC  :  cot.  BC. 

THEOREM  IY. 

(217.)  If  from  an  angle  of  a  spherical  triangle  a  perpen- 
dicular be  drawn  to  the  base,  then  the  tangent  of  half  the  sum 
of  the  segments  of  the  base  is  to  the  tangent  of  half  the  sum 
of  the  sides,  as  the  tangent  of  half  the  difference  of  the  sides 
is  to  the  tangent  of  half  the  difference  of  the  segments  of  the 
base. 

Let  ABC  be  any  spherical  trian- 
gle, and  let  CD  be  drawn  from  C 
perpendicular  to  the  base  AB  ;  then 
tan.  i(BD-f  AD) :  tan.  i(BC+AC) : : 
tan.  J(BC-AC)  :  tanf  l(BD-AD). 

Let  BC=a,  AC=£,  BD=m,  and 
AD^ra.  Then,  by  Theorem  III.,  Cor.  1, 

cos.  a  :  cos.  b  :  :  cos.  m  :  cos.  n 


SPHERICAL   TRIGONOMETRY.  165 

Whence,  Geom.,  Prop.  7,  Cor.,  B.  IT., 
cos.  #-f  co3.  a  :  cos.  b— cos.  a  : :  cos.  ra+cos.  m  :  cos.  w—cos.  m 

But  by  Trig.,  Art.  76, 

cos.  &+COS.  a  :  cos.  6— cos.  a  :  :  cot.  \(a-\-b)  :  tan.  ^(a—b). 

Also,  by  the  same  Art., 
cos.  w+cos.  m  :  cos.  n— cos.  m  : :  cot.  ^(m+n)  :  tan.  J(wi— n) 

Therefore 

cot.  ^(a+b)  :  cot.  \(m-\-ri)  :  :  tan.  \(a— b)  :  tan.  ^(m—n). 

But,  since  tangents  are  reciprocally  as  their  cotangents,  Art, 
28,  we  have 

cot.  \(a+b)  :  cot.  \(m-\-ri)  :  :  tan.  \(m-\-ri)  :  tan.  J(#+&) 

Hence 

tan.  \(m+n)  :  tan.  ^(a+b)  :  :  tan.  ^(#— b)  :  tan.  ^(w— w). 

(218.)  In  the  solution  of  oblique-angled  spherical  triangle* 
six  cases  may  occur,  viz. : 

1.  Griven  two  sides  and  an  angle  opposite  one  of  them. 

2.  Griven  two  angles  and  a  side  opposite  one  of  them. 

3.  Griven  two  sides  and  the  included  angle. 

4.  Griven  two  angles  and  the  included  side. 

5.  Griven  the  three  sides. 

6.  Griven  the  three  angles. 

CASE  I. 

(219.)  Given  two  sides  and  an  angle  opposite  one  of  them^ 
to  find  the  remaining  parts. 

In  the  triangle  ABC,  let  there  be  given  the  two  sides  AC 
and  BC,  and  the  angle  A  opposite  one 
of  them.    The  angle  B  may  be  found 
by  Theorem  III. 
sin.  BC  :  sin.  AC  :  :  sin.  A  :  sin.  B. 

From  the  angle  C  let  fall  the  per- 
pendicular  CD  upon  the  side  AB. 
The  triangle  ABC  is  divided  into  two  right-angled  triangles, 
in  each  of  which  there  is  given  the  hypothenuse  and  the  anglo 
at  the  base.  The  remaining  parts  may  then  be  found  by  Na- 
pier's rule. 

Ex.  1.  In  the  oblique-angled  spherical  triangle  ABC,  the 


I  fit)  TRIGONOMETRY. 

side  AC=70°  10'  30",  BC=80°  5'  4",  and  the  angle  A=33C 
15'  7".     Required  the  other  parts. 

sin  BC  :  sin.  AC  :  :  sin.  A  :  sin.  B=31°  34'  38" 
Then,  in  the  triangle  ACD, 

R  cos.  AC = cot.  -A  cot.  ACD. 
Whence  ACD=77°  27'  47". 

Also,  in  the  triangle  BCD, 

R  cos.  BC=cot.  B  cot.  BCD. 
Whence  BCD=  83°  57'  29". 

Therefore  ACB=161°  25'  16". 

To  find  the  side  AB. 

sin.  A  :  sin.  ACB  : :  sin.  BC  :  sin.  AB=145°  5'  0". 

When  we  have  given  two  sides  and  an  opposite  angle,  there 
are,  in  general,  two  solutions,  each  of  which  will  satisfy  the 
conditions  of  the  problem.  If  the  side  AC,  the  angle  A,  and 
the  side  opposite  this  angle  are  given, 
then,  with  the  latter  for  radius,  de- 
scribe an  arc  cutting  the  arc  AB  in 
the  points  B  and  B'.  The  arcs  CB, 
CB'  will  be  equal,  and  each  of  the  tri- 
angles ACB,  ACB'  will  satisfy  the 
conditions  of  the  problem.  There  is  the  same  ambiguity  in 
the  numerical  computation.  The  angle  B  is  found  by  means 
of  its  sine.  But  this  may  be  the  sine  either  of  ABC,  or  of  its 
supplement  AB'C  (Art.  27).  In  the  preceding  example,  the 
first  proportion  leaves  it  ambiguous  whether  the  angle  B  is 
31°  34'  38',  or  its  supplement  148°  25'  22".  In  order  to  avoid 
false  solutions,  we  should  remember  that  the  greater  side  of 
a  spherical  triangle  must  lie  opposite  the  greater  angle,  and 
conversely  (Geom.,  Prop.  17,  B.  IX.).  Thus,  since  in  the  pre- 
ceding example  the  side  AC  is  less  than  BC,  the  angle  B  must 
be  less  than  A,  and,  therefore,  can  not  be  obtuse. 

If  the  quantity  sought  is  determined  by  means  of  its  cosine, 
tangent,  or  cotangent,  the  algebraic  sign  of  the  result  will 
show  whether  this  quantity  is  less  or  greater  than  90° ;  for  the 
cosines,  tangents,  and  cotangents  are  positive  in  the  first  quad- 
rant, and  negative  in  the  second.  Hence  the  algebraic  sign 


SPHERICAL    TRIGONOMETRY.  Ib? 

of  each  term  of  a  proportion  should  be  preserved  whenever  one 
of  them  is  negative. 

Ex.  2.  In  the  spherical  triangle  ABC,  the  side  a =124°  53', 
6=31°  19',  and  the  angle  A=16°  26'.  Required  the  remain- 
ing  parts. 

(  B=  10°  19'  34' 
Ans.  }  0  =  171°  48'  22" 
(  c  =155°  35'  22' . 

CASE  II. 

(220.)  Given  two  angles  and  a  side  opposite  one  of  them, 
to  find  the  remaining  parts. 

In  the  triangle  ABC  let  there  be  given  two  angles,  as  A  and 
B,  and  the  side  AC  opposite  to  one 
of  them.  The  side  BC  may  be 
found  by  Theorem  III. 
sin.  B  :  sin.  A  : :  sin.  AC  :  sin.  BC. 
From  the  unknown  angle  C  draw 
CD  perpendicular  to  AB  ;  then  will 
the  triangle  ABC  be  divided  into  two  right-angled  triangles,  in 
each  of  which  there  is  given  the  hypothenuse  and  the  angle  at 
the  base.  "Whence  we  may  proceed  by  Napier's  rule,  as  in 
Case  I. 

Ex.  1.  In  the  oblique-angled  spherical  triangle  A.BC,  there 
are  given  the  angle  A=52°  20',  B=63°  40',  and  the  sido 
fi  =  83°  25'.  Required  the  remaining  parts. 

sin.  B  :  sin.  A  :  :  sin.  AC  :  sin.  BC  =  61°  19'  53". 
Then,  in  the  triangle  ACD, 

cot.  AC  :  R  :  :  cos.  A  :  tan.  AD =79°  18'  17". 
Also,  in  the  triangle  BCD, 

cot.  BC  :  R  :  :  cos.  B  :  tan.  BJt)=39°  3'  8". 
Hence  AB-1180  21'  25". 

To  find  the  angle  ACB. 
sin.  BC  :  sin.  AB  :  :  sin.  A  :  sin.  ACB=127°  26'  47". 

When  we  have  given  twc  angles  and  an  opposite  side,  there 
are,  in  general,  two  solutions,  each  of  which  will  satisfy  the 
conditions  of  the  problem.  If  the  angle  A,  the  side  AC,  and 


168  TRIGONOMETRY. 

the  angle  opposite  this  side  are  given,  then  through  the  point 
C  there  may  generally  be 
drawn  two  arcs  of  great  cir- 
cles CB,  CB',  making  the 
same  angle  with  AB,  and 
each  of  the  triangles  ABC, 
AB'C  will  satisfy  the  condi- 
tions of  the  problem.  There  is  the  same  ambiguity  in  the 
numerical  computation,  since  the  side  BC  is  found  by  means 
of  its  sine  (Art.  27).  In  the  preceding  example,  however, 
there  is  no  ambiguity,  because  the  angle  A  is  less  than  B, 
and,  therefore,  the  side  a  must  be  less  than  #,  that  is,  less  than 
a  quadrant. 

Ex.  2.  In  the  oblique-angled  spherical  triangle  ABC,  the 
angle  A  is  128°  45',  the  angle  0=30°  35',  and  BC  =  68°50'. 
Required  the  remaining  parts. 

It  will  be  observed  that  in  this  case  the  perpendicular  BD, 
drawn  from  the  angle  B,  falls  without  the  triangle  ABC,  and 
therefore  the  side  AC  is  the  difference  between  the  segimrtf* 
CD  and  AD.  (  AB=37°  28'  20  ' 

Am.  ]  AC=40°    9'    4". 
(  B    -32°  37'  58" 

CASE  III. 

(221.)  Given  two  sides  and  the  included  angle,  to  find  th 
remaining'  parts. 

In  the  triangle  ABC  let  there  be  given  two  sides,  as  AE 
AC,  and  the  included  angle  A.     Let 
fall  the  perpendicular  CD  on  the  side 
AB  ;  then,  by  Napier's  rule, 

R  cos.  A=tan.  AD  cot.  AC. 

Having  found  the  segment  AD,  the 
segment  BD  becomes  known;  then, 
by  Theorem  III.,  Cor.  3, 

sin.  BD  :  sin.  AD  :  :  tan.  A  :  tan.  B. 

The  remaining  parts  may  now  be  found  by  Theorem  III. 

Ex.  1.  In  the  spherical  triangle  ABC,  the  side  AB=73°  20 , 
AC=41°  45',  and  the  angle  A=30°  30'.  Required  tho  remain- 
ing  parts. 


SPHERICAL   TRIGONOMETRY.  1()9 

cot.  AC  :  cos.  A  : :  R  :  tan,  AD=37°  33'  41". 
Hence  BD=35°  46'  19". 

sin.  BD  :  sin.  AD  : :  tan.  A.  :  tan.  B=31°  33'  43". 
Also,  "by  Theorem  III.,  Cor.  1, 

cos.  AD  :  cos.  BD  :  :  cos.  AC  :  cos.  BC=403  33'  0". 
Then,  by  Theorem  III., 

sin.  BC  :  sin.  AB  : :  sin.  A  :  sin.  ACB=131°  6'  47' . 
Ex.  2.  In  the  spherical  triangle  ABC,  the  side  AB=78°  15', 
AC=56°  20',  and  the  angle  A=120°.      Required  the  other 
parts. 

(  B   =48°  57'  29". 

Ans.  }  C    =62°  31'  40". 

(BC-1070    7' 45" 

CASE  IY. 

(222.)  Given  two  angles  and  the  included  side,  to  find  tne 
remaining  parts. 

In  the  triangle  ABC  let  there  be  given  two  angles,  as  A  and 
ACB,  and  the  side  AC  included  be- 
tween them.     From  C  let  fall  the  per- 
pendicular CD  on  the  side  AB.     Then, 
by  Napier's  rule, 

R  cos.  AC=cot.  A  cot.  ACD. 
Having  found  the  angle  ACD,  the 
angle  BCD  becomes  known ;  then,  by 
Theorem  III.,  Cor.  4, 

cos.  ACD  :  cos.  BCD  :  :  cot.  AC  :  cot.  BC. 
The  remaining  parts  may  now  be  found  by  Theorem  Hi. 
Ex.  1.  In  the  spherical  triangle  ABC,  the  angle  A=32°  10', 
the  angle  ACB =133°  20',  and  the  side  AC =39°  15'.     Re. 
quired  the  other  parts. 

cot.  A  :  cos.  AC  :  :  R  :  cot.  ACD=64°  1'  57". 
Hence  BCD=69°  18'  3". 

Then 

cos.  ACD  :  cos.  BCD  :  :  cot.  AC  :  cot.  BC=45°  20'  43' 
Also,  by  Theorem  III.,  Cor.  2, 

sin.  ACD  :  sin.  BCD  : :  cos.  A  :  cos.  B=28°  15'  47". 
Then,  by  Theorem  III., 

sin.  B  :  sin  ACB  :  :  sin.  AC  :  sin.  AB-r-760  23'  ,7'. 


170  TRIGONOMETRY. 

Ex.  2.  In  the  spherical  triangle  ABC,  the  angle  A=125°  20'. 
the  angle  C=4S'J  30',  and  the  side  AC =83°  13'.  ReauireJ 
•lie  remaining  parts. 

(  AB=  56°  39'    9". 

Ans.  }  BC  =  114°  30'  24". 

(  B    =62°  54'  38  . 

CASE  V. 

(223.)  Given  the  three  sides  of  a  spherical  triangle,  to  find 
the  angles. 

In  the  triangle  ABC  let  there  be  given  the  three  sides. 
From  one  of  the  angles,  as  C,  draw  CD 
perpendicular  to  AB.     Then,  by  The- 
orem IV.,  tan.  JAB  :  tan.  J(AC+BC)  : 
tan.  J(AC-BC)  :  tan.  J(AD-BD). 

Hence  AD  and  BD  become  known ; 
then,  by  Napier's  rule, 

R  cos.  A=tan.  AD  cot.  AC. 
The  other  angles  may  now  be  easily  found. 
It  is  generally  most  convenient  to  let  fall  the  perpendiculai 
upon  the  longest  side  of  the  triangle. 

Ex.  1.  In  the  spherical  triangle  ABC,  the  side  AB=1123 
25',  AC =60°  20',  and  BC=81°  10'.     Required  the  angles, 
tan.  56°  12J'  :  tan.  70°  45'  :  :  tan.  10°  25'  :  tan.  19°  24'  26". 
Hence     AD=36°  48'  4",  and  BD=75°  36'  56". 
Then    R  :  tan.  AD  :  :  cot.  AC  :  cos.  A=64°  46'  36". 
Also,     R  :  tan.  BD  :  :  cot.  BC  :  cos.  B=52°  42'  12". 
Then  sin.  AC  :  sin.  AB  :  :  sin.  B  :  sin.  ACB=122°  IV  6". 
Ex.  2.  In  the  spherical  triangle  ABC,  the  side  AB=40°  35' 
AC -39°  10',  and  BC  =  7l°  15  .     Required  the  angles. 

(  A=130°  35'  55". 

Ans.  ]E=  30°  25'  34" 

(  C-  31°  26'  32". 

CASE  VI. 

(224.)  Given  the  three  angles  of  a  spherical  triangle,  to 
find  the  sides. 

If  A,  B,  C  are  the  angles  of  the  given  triangle,  and  a,  b<  c 
its  sides,  then  1SO°~A,  180°-B,  and  1SO°-C  are  the  side* 


SPHERICAL    TRIGONOMETRY. 


171 


ol  its  polar  triangle,  whose  angles  may  be  found  by  Case  "V 
Then  the  supplements  of  those  angles  will  be  the  sides  a,  Z>,  t 
of  the  proposed  triangle. 

Ex.  1.  In  the  spherical  triangle  ABC,  the  angle  A=125C 
3-4',  B=98°  44',  and  C=61°  53'.     Required  the  sides. 

The  sides  of  the  polar  triangle  are 

54°  26',  81°  16',  and  118°  7'. 

From  which,  by  Case  V.,  the  angles  are  found  to  be 
134°  6'  21",  41°  28'  17",  and  53°  34'  47". 

Hence  the  sides  of  the  proposed  triangle  are 
AB=45°  53'  39"   BC=138°  31'  43"   and  AC=126°  25'  13' 


Ex.  2.  In  the  spherical  triangle  ABC,  the  angle  A= 
55',  B  =  116°  38',  and  C  =  120°  43'.     Required  the  sides. 

(  a=  98°  21'  20". 
Ans.  ]b  =  W9°  50'  10". 
13'    7" 


TRIGONOMETRICAL  FORMULAE. 

(225.)  Let  ABC  be  any  spherical 
triangle,  and  from  the  angle  B  draw 
the  arc  BD  perpendicular  to  the  base 
AC.  Represent  the  sides  of  the  trian- 
gle by  &,  b)  c,  and  the  segment  AD  by 
x  ;  then  will  CD  be  equal  to  b—x. 

By  Theorem  III.,  Cor.,  1, 
cos.  c  :  cos.  a  :  :  cos.  x  :  cos.  (b—x) 

cos.  b  cos.  a;+sin.  b  sin.  x 
:  :  cos.  x  :  P 

(Trig.,  Art.  72),  formula  (4). 

Whence 

R  cos.  a  cos.  a;— cos.  b  cos.  c  cos.  £+sin.  b  cos.  c  sin.  x, 
or,  dividing  each  term  by  cos.  #,  and  substituting  the  value  <i' 

— • —  (Art.  28 \  we  obtain 
cos.  x  v 

R2  cos.  a=R  cos.  b  cos.  £-{-sin.  b  cos.  c  tan.  x 
But  by  Theorem  II.,  Cor.  2,  we  have 

R  cos.  A    cos.  A  sin.  c  t 

(Art,  28). 

cot.  c  cos.  c. 


tan.  x= 


172  TRIGONOMETRV. 

Hence   Ra  cos.  a=IL  cos.  b  cos.  c+sin.  b  sin.  c  cos.  A,    (1) 
from  which  all  the  formulae  necessary  for  the  solution  of  spheri- 
cal triangles  may  be  deduced. 
In  a  similar  manner  we  obtain 

Ra  cos.  &=R  cos.  a  cos.  c+sin.  a  sin.  c  cos.  B,         (2) 
Ra  cos.  c=R  cos.  a  cos.  £+sin.  a  sin.  b  cos.  C.         (3) 
These  equations  express  the  following  Theorem : 
The  square  of  radius  multiplied  by  the  cosine  of  either  side 
of  a  spherical  triangle,  is  equal  to  radius  into  the  product  of 
the  cosines  of  the  two  other  sides,  plus  the  product  of  the  sines 
of  those  sides  into  the  cosine  of  their  included  angle. 
(226.)  From  equation  (1)  we  obtain,  by  transposition, 
Ra  cos.  a— R  cos.  b  cos.  c 

COS.  A  = : — : , 

sm.  b  sm.  c 

a  formula  which  furnishes  an  angle  of  a  triangle  when  the 
three  sides  are  known. 

If  we  add  R  to  each  member  of  this  equation,  we  shall  have 
Ra  cos.  &+R  sin.  b  sin.  c— R  cos.  b  cos.  c 


R+cos.  A=: 


sin.  b  sin.  c 
2  cos.  3£A 


But,  by  Art.  74,  R+cos.  A=- 

And,  by  Art.  72,  formula  (2),  by  transposition, 

R  sin.  b  sin.  c—  R  cos.  b  cos.  c=—  Ra  cos.  (b-\-c). 
Hence,  by  substitution,  we  obtain 

2  cos.  2|A_Ra(cos.  a-cos.  (b+c)) 
R  sin.  b  sin.  c 

_2R  sin.  %(a+b+c)  sin.  %(b+c—a) 

sin.  b  sin.  c 
by  Art.  75,  formula  (4). 

If,  then,  we  put  s=^(a+b+c),  that  is,  half  the  sum  of  the 
sides,  we  shall  find 


/sin.  s  sin.  (s—  a) 
=R  V  --  :  —  r—  ^  -  •• 


.          . 

cos.  AA=R  V  --  :  —  r—  ^  -  ••  (4; 

sm.  b  sm.  c 

By  subtracting  cos.  A  from  R  instead  of  adding,  we  slial 
obtain,  in  a  similar  manner, 

_.     /sin.  (s—  b)  sin.  (s  —  c)  . 

sm.  JA=R  V  --  ^  —  r—  =—      —  -•          (5) 

v  sm.  b  sm.  c 


SPHERICAL    TRIGONOMETRY.  173 

Either  formula  (4)  or  (5)  may  be  employed  to  compute  the 
angles  of  a  spherical  triangle  when  the  three  sides  are  known, 
and  this  method  may  be  preferred  to  that  of  Art.  223. 

Ex.  1.  In  a  spherical  triangle  there  are  given  a=63°  50'. 
ft =80°  19',  and  c=120°  47'.     Requirpd  the  three  angles. 
Here  half  the  sum  of  the  sides  is  132°  28'=*. 
Also,  s-a=68°  38'. 

Using  formula  (4),  we  have 

log.  sine  s,  132°  28'  .  .  9.867862 
log.  sine  (s-a),  68°  38'  .  .  9.969075 
-log.  sine  £,  80°  19'  comp.  0.006232 
-log.  sine  c,  120°  47'  comp.  0.065952 

Sum     19.909121 

log.  cos.  JA,          25°  45'  19"        9.954560. 
Hence  the  angle  A=51°  30'  38". 

The  remaining  angles  may  be  found  by  Theorem  TIL,  or  by 
formulas  similar  to  formula  (4). 


sm.  a  sin.  c 


/sin. 
.  iQ-BV 


s  sin.  (s—  c) 
cos. 


•      /     • 

sm.  a  sin.  b 

We  thus  find  the  angle  B=  59°  16'  46", 
and  0=131°  28'  36". 

Ex.  2.  In  a  spherical  triangle  there  are  given  a=115°  20'? 
£=57°  30',  and  c-82°  28'.  Required  the  three  angles. 

(  A-  126°  35'    2". 

Ans.  ]  B=  48°  31'  42". 

(  C=  61°  43'  58". 

(227.)  By  means  of  the  polar  triangle,  we  may  convert  the 
preceding  formulae  for  angles  into  formulae  for  the  sides  of  a 
triangle,  since  the  angles  of  every  triangle  are  the  supplements 
of  the  sides  of  its  polar  triangle.  Let,  then,  a',  £',  c',  A',  B', 
C'  represent  the  sides  and  angles  of  the  polar  triangle,  and  we 
shall  have 

A=1803-a',  B=180°-6',  C=180°-c", 
a=180°-A;,  &=180°-B;,  c=180°-C' 
Therefore        sin.  |A=sin.  (90°-^')  =cos.  J«'t 
cos.  |A=cos.  (90°—  fa')  =sin.  |a', 


L74  TRIGONOMETRY. 

sin.  b=x'm.  V180°-B')  =  sm.  B', 
sin.  c=sin.  (180°  — C')=sin.  C'. 

Also,  il  we  put  S'=half  the  sum  of  the  angles  of  the  poiai 
triangle,  we  shall  have 


or  s=270°-S', 

whence  sin.  s=—  cos.  S', 

sin.  (s-a)=sm.  [90°-(S'-A')]=cos.  (S'-A'^ 

sin.  (s-b)=cos.  (S'-B'), 

sin.  (s-c)=cos.  (S'-C'). 

By  substituting  these  values  in  formula  (5),  Art.  226,  and 
omitting  all  the  accents,  since  the  equations  are  applicable  to 
any  triangle,  we  obtain 


.       _    /cos.  (S-B)cos.(S-C) 
cos.  Ja=Ry  -  *-.  —  ^—.  —  ^  -  ';  6) 

sin.  B  sin.  C 

and  formula  (4)  becomes 


.. 

sin.  *O=EV-  —  -  —  p   •  V.  —  -j  (7) 

sin.  B  sm.  C 

which  formulae  enable  us  to  compute  the  sides  of  a  triangle 
when  the  three  angles  are  known;  and  this  method  may  bo 
preferred  to  that  of  Art.  224. 

In  a  similar  manner,  by  means  of  the  polar  triangle,  wo 
derive  from  formula  (1),  Art.  225,  the  equation 

R3  cos.  A=cos.  a  sin.  B  sin.  C—  R  cos.  B  cos.  C  ;     (8) 

that  is,  the  square  of  radius  multiplied  by  the  cosine  of  either 
angle  of  a  spherical  triangle,  is  equal  to  the  product  of  the 
sines  of  the  two  oilier  angles  into  the  cosine,  of  their  included 
tide,  minus  radius  into  the  product  of  their  cosines. 

Ex.  1.  In  a  spherical  triangle  ABC,  there  are  given  A— 
130°  30',  B=30°  50',  and  0=32°  5'.  Required  the  three 
sides. 

Here  half  the  sum  of  the  angles  is  96°  42  30V=S. 

Also,  S-A=-33°47'  30", 

S-B=     65°  52'  30", 
S-C=     64°  37'  30" 

Using  formula  (6),  we  have 


SPHERICAL   TRIGONOMETRY.  175 

log.  cos.  (S-B),  65°  52'  30"  .  9.611435 
log.  cos.  (S-C),  64°  37'  30"  .  9.631991; 
-log.  sin.  B,-  30°  50'  comp.  0.290270 
-log.  sin.  C,  32°  5'  comp.  0.274781 

Sum  19.808478 

log.  cos.  \a,        36°  40'  1"  9.904239. 

Hence  the  side  a= 73°  20'  2". 

The  remaining  sides  may  be  found  by  Theorem  TIL,  or  by 
formulas  similar  to  formula  (6). 

4  /cos.  (S-A)cos.  (S-C) 

cos.  i£=R\/ ^ r— - — ~-  -, 

sin.  A  sm.  C 


.       _     /cos.  (S-A)cos.  (S-B) 

COS.  ^C  =  li\/  : T : ^ . 

sin.  A  sm.  1> 

We  thus  find  the  side  £=40°  13°  12", 
and  c-42°    0'  12". 

Ex.  2.  In  the  spherical  triangle  ABC,  the  angle  A=129: 
30',  B=54°  35',  and  C  =  63°  5'.  Required  the  three  sides. 

(  a=120°  57'    5". 

Ans.  }b=  64°  55'  37" 

(c=  82°  19'    0  '. 

(228.)  Formula  (1),  Art.  225,  will  also  furnish  a  new  test 
for  removing  the  ambiguity  of  the  solution  in  Case  I.  of  oblique 
angled  triangles.     For  we  have 

R2  cos.  a— R  cos.  b  cos.  c 

COS.   A  = : 7 : . 

sm.  b  sm.  c 
Now  if  cos.  a  is  greater  than  cos.  #,  we  shall  have 

R2  cos.  &>R  cos.  b  cos.  c, 

or  the  sign  of  the  second  member  of  the  equation  will  be  the 
same  as  that  of  cos.  a,  since  the  denominator  is  necessarily 
positive,  and  cos.  c  is  less  than  radius.  Hence  cos.  A  and  cos. 
a  will  have  the  same  sign ;  or  A  and  a  will  be  of  the  same 
species  when  cos  &>cos.  £,  or  sin.  #<sin.  b ;  that  is, 

If  the  sine  of  the  side  opposite  to  the  required  angle  is  less 
than  the  sine  of  the  other  given  side,  there  will  be  but  one 
triangle. 

But  if  cos.  a  is  less  than  cos.  Z>,  then  such  a  value  may  be 
given  to  c  as  to  render 


176  TRIGONOMETRY. 

R2  cos.  &<R  cos.  b  cos.  c, 

or  the  sign  of  the  second  member  of  the  equation  will  depend 
upon  the  value  of  cos.  c  ;  that  is,  c  may  be  taken  so  as  to  ren- 
der cos.  A  either  positive  or  negative.  Hence 

If  the  sine  of  the  side  opposite  to  the  required  angle  is 
greater  than  the  sine  of  the  other  given  side,  there  will  be 
two  triangles  which  fulfill  the  given  conditions. 

(229.)  Formula  (8),  Art.  227,  will  furnish  a  trst  for  re- 
moving the  ambiguity  in  Case  II.  of  oblique-angled  triangles. 
For  we  have 

R2  cos.  A+R  cos.  B  cos.  C 

COS.  a— : = — : pq ; 

sin.  B  sin.  C 

trom  which  it  follows,  as  in  the  preceding  article,  that  if  cos.  A 
is  greater  than  cos.  B,  A  and  a  will  be  of  the  same  species.  But 
if  cos.  A  is  less  than  cos.  B,  then  such  values  may  be  given 
to  C  as  to  render  cos.  a  either  positive  or  negative.  Hence 

If  the  sine  of  the  angle  opposite  to  the  required  side  is  less 
than  the  sine  of  the  other  given  angle,  there  will  be  but  one 
triangle  ; 

But,  if  the  sine  of  the  angle  opposite  to  the  required  side 
is  greater  than  the  sine  of  the  other  given  angle,  there  will 
be  two  triangles  which  fulfill  the  given  conditions. 

SAILING  ON  AN  ARC  OF  A  GREAT  CIRCLE. 

(230.)  It  is  demonstrated  in  Geom.,  Prop.  3,  B.  IX.,  that  the 
shortest  path  from  one  point  to  another  on  the  surface  of  a 
sphere  is  the  arc  of  a  great  circle  which  joins  the  two  given 
points.  Hence,  if  it  is  desired  to  sail  from  one  port  to  another 
by  the  shortest  route,  it  is  necessary  to  follow  an  arc  of  a  great 
circle,  and  this  arc  generally  does  not  coincide  with  a  rhumb 
line. 

The  bearing  and  distance  from  one  place  to  another  on  the 
arc  of  a  great  circle  may  be  computed  from  the  latitudes  and 
longitudes  of  the  places  by  means  of  Spherical  Trigonometry 

Thus,  let  P  be  the  pole  of  the  earth,  EQ,  a  part  of  the  equa- 
tor, and  A  and  B  the  two  given  places  comprehended  between 
the  meridians  PE  and  PQ,.  Then  PA  is  the  complement  of 
the  latitude  of  A,  PB  is  the  complement  of  the  latitude  of  B, 


SPHERICAL   TRIGONOMETRY. 


1/7 


and  the  angle  P  is  measured  by  the  arc  EQ,,  which  is  the 
difference  of  longitude  between  the  two 
places.  Hence,  in  the  triangle  ABP,  we 
have  given  two  sides  AP,  BP,  and  the  in- 
cluded angle  P,  from  which  we  may  com- 
pute the  side  AB,  and  the  angles  A  and  B, 
according  to  Case  III.  of  oblique-angled  tri- 
angles. 

Ex.  1.  Required  the  course  and  distance 
from  Nantucket  Shoals,  in  latitude  41°  4' 
N.,  longitude  69°  55'  W.,  to  Cape  Clear,  in 
latitude  51°  26'  N.,  longitude  9°  29'  W.,  on  the  arc  of  a  gr«at 
circle. 

Here  we  have  given 

the  angle  P=69°  55'-  9°  29' =60°  26' ; 
the  side  PA=90°       -41°    4'=48°  56' ; 
the  side  PB=90°       -51°  26'=3S°  34'. 
Then  cot.  PB  :  cos.  P  : :  R  :  tan.  PD=21°  28'  35". 
Whence  AD=27°  27'  25". 

Also  sin.  AD  :  sin.  PD  :  :  tan.  P  :  tan.  A   =54°  27'  21  , 
and        sin.  A     :  sin.  PB  :  :  sin.  P  :  sin.  AB=41°  47'  28' . 
41°  47'  28"  is  equal  to  2507.47  nautical  miles. 
Hence  the  course  from  Nantucket 
Shoals  to  Cape  Clear  is  N.  54°  27' 
E.,  and  the  distance  is  2507.47  miles. 
According  to  Mercator's   sailing, 
the  course  on  a  rhumb-line,  found  on 
page  152,  is  N.  76°  E.,  and  the  dis- 
tance 2572.9  miles.     Hence  the  dis- 
tance on  an  arc  of  a  great  urclo  is  65.4  miles  less  than  on  a 
rhumb-line,  and  the  formei  course  is  21 J  degrees  more  north- 
erly than  the  latter. 

While  sailing  on  a  rhumb-line  the  course  of  a  ship  remains 
always  the  lame,  but  while  sailing  on  an  arc  of  a  great  circle 
the  course  is  continually  changing.  The  preceding  course  is 
lhat  with  which  the  ship  starts  from  Nantucket,  and  a  new 
computation  of  the  course  should  be  made  every  day  or  two ;  or 
.t  might  be  mor.e  convenient  to  compute  beforehand  the  position 
of  the  points  in  which  the  great  circle  intersects  the  meridians 

M 


178  TRIGONOMETRY. 

for  every  five  degrees  of  longitude,  and  the  ship  might   b* 
steered  upon  a  direct  course  for  these  points  successively. 

Ex.  2.  Required  the  course  and  distance  from  Nantuckel 
Shoals  to  G-ihraltar,  in  latitude  36°  6'  N.,  longitude  5°  2f  w  , 
on  the  shortest  route. 

Ans.  The  course  is  N.  73°  29'  E 

Distance  2974.1  miles. 

Ex.  3.  Required  the  course  and  distance  from  Sandy  Hook, 
in  latitude  40°  28'  N.,  longitude  74°  1'  "W.,  to  Madeira,  in 
latitude  32°  38'  N.,  longitude  16°  55'  "W.,  on  the  shortest  route. 

Ans.  The  course  is  N.  80°  53'  E. 

Distance  2744.1  miles. 

Ex.  4.  Required  the  course  and  distance  from  Sandy  Hook 
to  St.  Jago,  in  latitude  14°  54'  N.,  longitude  23°  30'  "W.,  on 
the  shortest  route. 

Ans.  The  course  is  S.  74°  46'  E. 

Distance  3037.6  miles. 

Ex.  5.  Required  the  course  and  distance  from  Sandy  Hook 
to  the  Cape  of  Good  Hope,  in  latitude  34°  22'  S.,  longitud« 
18°  30'  E.,  on  the  shortest  route. 

Ans.  The  course  is  S.  63°  48'  E 
Distance  6792  miles. 


EXAMPLES  FOR  PEACTICE. 

PLANE  TRIGONOMETRY. 

Prob.  1.  Given  the  three  sides  of  a  triangle,  627,  718.9,  and 
1140,  to  find  the  angles. 

Ans.  29°  44X  2",  34°  39'  26",  and  115°  36'  32". 
Prob.  2.  In  the  triangle  ABC,  the  angle  A  is  given  89°  45' 
43",  the  side  AB  654,  and  the  side  AC  460,  to  find  the  remain- 
ing parts. 

Ans.  BC  =  798;  the  angle  B  =  35°  12X  1",  and  the  angle 

C  =  55°  2/16//. 

Prob.  3.  In  the  triangle  ABC,  the  angle  A  is  given  56°  12" 
45",  the  side  BC  2597.84,  and  the  side  AC  3084.33,  to  find  the 
remaining  parts. 

Ans.  B  =  80°  39'  40",  C  =  43°    7'  35",  c  =  2136.8 ; 
or,  B  =  99    2020,    C  =  24    2655,    c  =  1293.8. 
Prob.  4.  In  the  triangle  ABC,  the  angle  A  is  given  44°  13' 
24",  the  angle  B  55°  59'  58".  and  the  side  AC  368,  to  find  the 
remaining  parts. 

Ans.  C  =  79°  46'  38",  AB  =  436.844,  and  BC  =  309.595. 
Prob.  5.  In  a  right-angled  triangle,  if  the  surn  of  the  hy- 
pothenuse  and  base  be  3409  feet,  and  the  angle  at  the  base  53° 
12'  14",  what  is  the  perpendicular  ? 

Ans.  1707.2  feet. 

Prob.  6.  In  a  right-angled  triangle,  if  the  difference  of  the 
hypothenuse  and  base  be  169.9  yards,  and  the  angle  at  the  base 
42°  36'  12",  what  is  the  length  of  the  perpendicular  ? 

Ans.  435.732  yards. 

Prob.  7.  In  a  right-angled  triangle,  if  the  sum  of  the  base 
and  perpendicular  be  123.7  feet,  and  the  angle  at  the  base  58° 
19'  32",  what  is  the  length  of  the  hypothenuse  ? 

Ans.  89.889  feet. 

Prob.  8.  In  a  right-angled  triangle,  if  the  difference  of  the 
base  and  perpendicular  be  12  yards,  and  the  angle  at  the  base 
38°  V  8",  what  is  the  length  of  the  hypothenuse  ? 

Ans.  69.81  yards. 


180  TRIGONOMETRY. 

Prob.  9.  A  May-pole,  50  feet  11  inches  high,  at  a  certain 
time  will  cast  a  shadow  98  feet  6  inches  long ;  what,  then,  is 
the  breadth  of  a  river  which  runs  within  20  feet  6  inches  of  the 
foot  of  a  steeple  300  feet  8  inches  high,  if  the  steeple  at  the 
same  time  throws  its  shadow  30  feet  9  inches  beyond  the  stream  ? 

Ans.  530  feet  5  inches. 

Prob.  10.  A  ladder  40  feet  long  may  be  so  placed  that  it  shall 
reach  a  window  33  feet  from  the  ground  on  one  side  of  the 
street,  and  by  turning  it  over,  without  moving  the  foot  out  of 
its  place,  it  will  do  the  same  by  a  window  21  feet  high  on  the 
other  side.  Required  the  breadth  of  the  street. 

Ans.  56.649  feet. 

Prob.  11.  A  May-pole,  whose  top  was  broken  off  by  a  blast 
of  wrind,  struck  the  ground  at  the  distance  of  15  feet  from  the 
foot  of  the  pole ;  what  was  the  height  of  the  whole  May-pole, 
supposing  the  length  of  the  broken  piece  to  be  39  feet  ? 

Ans.  75  feet. 

'Prob.  12.  How  must  three  trees,  A,  B,  C,  be  planted,  so  that 
the  angle  at  A  may  be  double  the  angle  at  B,  the  angle  at  B 
double  the  angle  at  C,  and  a  line  of  400  yards  may  just  go 
round  them  ? 

Ans.  AB  =  79.225,  AC  =  142.758,  and  B  0  =  178.017  yards. 

Prob.  13.  The  town  B  is  half  way  between  the  towns  A  and 
C,  and  the  towns  B,  C,  and  D  are  equidistant  from  each  other. 
What  is  the  ratio  of  the  distance  AB  to  AD  ? 

Ans.  As  unity  to  -v/3. 

Prob.  14.  There  are  two  columns  left  standing  upright  in  the 
ruins  of  Persepolis ;  the  one  is  66  feet  above  the  plain,  and  the 
other  48.  In  a  straight  line  between  them  stands  an  ancient 
statue,  the  head  of  which  is  100  feet  from  the  summit  of  the 
higher,  and  84  feet  from  the  top  of  the  lower  column,  the  base 
of  which  measures  just  74  feet  to  the  centre  of  the  figure's  base. 
Required  the  distance  between  the  tops  of  the  two  columns. 

Ans.  156.68  feet. 

Prob.  15.  Prove  that  tang.  (45°-£)=^       ~ L|- 

Prob.  16.  One  angle  of  a  triangle  is  45°,  and  the  perpendic- 
ular from  this  angle  upon  the  opposite  base  divides  the  base  into 
two  parts,  which  are  in  the  ratio  of  2  to  3.  "What  are  the 


EXAMPLES   FOR   PRACTICE.  181 

parts  into  which  the  vertical  angle  is  divided  by  this  perpen- 
dicular ? 

Ans.  18°  26'  6"  and  26°  33'  54". 
Prob.  17.  Prove  that  sin.  36  =  3  sin.  &-4  sin.3  b. 
Prob.  18.  One* side  of  a  triangle  is  25,  another  is  22,  and  the 
angle  contained  by  these  two  sides  is  one  half  of  the  angle  op- 
posite the  side  25.     What  is  the  value  of  the  included  angle? 

Ans.  39°  58'  51". 

Prob.  19.  One  side  of  a  triangle  is  25,  another  is  22,  and 
the  angle  contained  by  these  two  sides  is  one  half  of  the  angle 
opposite  the  side  22.  What  is  the  value  of  the  included  angle  ? 

Ans.  30°  46'  38". 

Prob.  20.  Two  sides  of  a  triangle  are  in  the  ratio  of  11  to  9, 
and  the  opposite  angles  have  the  ratio  of  3  to  1.  What  are 
those  angles  ? 

Ans.  The  sine  of  the  smaller  of  the  two  angles  is  -|,  and  of 
the  greater  ff ;  the  angles  are  41°  48X  37"  and 
125°  25'  51". 

Prob.  21.  One  side  of  a  triangle  is  15,  and  the  difference  of 
the  two  other  sides  is  6 ;  also,  the  angle  included  between  the 
first  side  and  the  greater  of  the  two  others  is  60°.  What  is  the 
length  of  the  side  opposite  to  this  angle  ? 

Ans.  57. 

Prob.  22.  One  side  of  a  triangle  is  15,  and  the  difference  of 
the  two  other  sides  is  6 ;  also,  the  angle  opposite  to  the  greater 
of  the  two  latter  sides  is  60°.  What  is  the  length  of  said  side  ? 

Ans.  13. 

Prob.  23.  One  side  of  a  triangle  is  15,  and  the  opposite  an- 
gle is  60°  ;  also,  the  difference  of  the  two  other  sides  is  6 
What  are  the  lengths  of  those  sides  ? 

Ans.  11.0712  and  17.0712. 

Prob.  24.  The  perimeter  of  a  triangle  is  100  ;  the  perpendic- 
ular let  fall  from  one  of  the  angles  upon  the  opposite  base  is  30, 
and  the  angle  at  one  end  of  this  base  is  50°.  What  is  the 
length  of  the  base  ? 

Ans.  30.388 


182  TRIGONOMETRY. 

t 

MENSURATION  OF  SURFACES  AND  SOLIDS. 

Prob.  1.  The  base  of  a  triangle  is  20  feet,  and  its  altitude 
18  feet.  It  is  required  to  draw  a  line  parallel  to  the  base  so  as 
to  cut  off  a  trapezoid  containing  80  square  feet.  "What  is  the 
length  of  the  line  of  section,  and  its  distance  from  the  base  of 
the  triangle? 

Ans.  Length  14.907  feet ;  distance  from  base  4.584  feet. 

Prob.  2.  The  base  of  a  triangle  is  20  feet,  one  angle  at  the 
base  is  63°  26',  and  the  other  angle  at  the  base  is  56°  19X.  It 
is  required  to  draw  a  line  parallel  to  the  base,  so  as  to  cut  off  a 
trapezoid  containing  109  square  feet.  What  is  the  length  of 
the  line  of  section,  and  its  distance  from  the  base  of  the  tri- 
angle ? 

Ans.  Length  12.070  feet;  distance  from  base  6.797  feet. 

Prob.  3.  In  a  perpendicular  section  of  a  ditch,  the  breadth  at 
the  top  is  26  feet,  the  slopes  of  the  sides  are  each  45°,  and  the 
area  140  square  feet.  Required  the  breadth  at  bottom  and  the 
depth  of  the  ditch. 

Ans.  Breadth  10.77  feet ;  depth  7.615  feet. 

Prob.  4.  The  altitude  of  a  trapezoid  is  23  feet ;  the  two  par- 
allel sides  are  76  and  36  feet ;  it  is  required  to  draw  a  line  par- 
allel to  the  parallel  sides,  so  as  to  cut  off  from  the  smaller  end 
of  the  trapezoid  a  part  containing  560  square  feet.  What 
is  the  length  of  the  line  of  section,  and  its  distance  from  the 
shorter  of  the  two  parallel  sides  ? 

Ans.  Length  56.954  feet;  distance  12.048  feet. 

Prob.  5.  From  the  greater  end  of  a  trapezoidal  field  whose 
parallel  ends  and  breadth  measure  12,  8,  and  10^-  chains  re- 
spectively, it  is  required  to  cut  off  an  area  of  six  acres  by  a 
fence  parallel  to  the  parallel  sides  of  the  field.  What  is  the 
length  of  the  fence,  and  its  distance  from  the  greater  side. 

Ans.  Length  of  fence  9.914  chains ;  distance  from  greater 
side  5.476  chains. 

Prob.  6.  There  are  three  circles  whose  radii  are  20,  28,  and 
29  inches  respectively.  Required  the  radius  of  a  fourth  circle, 
whose  area  is  equal  to  the  sum  of  the  areas  of  the  other  three. 

Ans.  45  inches. 

Prob.  7.  In  constructing  a  rail-road,  the  pathway  of  which 


EXAMPLES   FOR   PRACTICE.  183 

is  24  feet  broad,  it  is  necessary  to  make  a  cutting  40  feet  in 
depth;  what  must  be  the  breadth  of  the  cutting  at  top,  sup- 
posing the  slopes  of  the  sides  to  be  65°  ? 

Ans.  61.305  feet. 

Prob.  8.  The  sides  of  a  quadrilateral  field  are  690  yards, 
467  yards,  359  yards,  and  428  yards ;  also,  the  angle  contained 
between  the  first  and  second  sides  is  57°  30',  and  the  angle  be- 
tween the  third  and  fourth  sides  122°  30'.  Required  the  area 
of  the  field. 

Ans.  200677.2  square  yards. 

Prob.  9.  There  are  two  regular  pentagons,  one  inscribed  in  a 
circle,  and  the  other  described  about  it ;  and  the  difference  of 
the  areas  of  the  pentagons  is  100  square  inches.  Required  the 
radius  of  the  circle. 

Ans.  8.926  inches. 

Prob.  10.  What  is  the. length  of  a  chord  cutting  off  one  third 
part  of  a  circle,  whose  diameter  is  289  feet. 

Ans.  278.67  feet. 

Prob.  11.  The  area  of  a  triangle  is  1012 ;  the  length  of  the 
side  a  is  to  that  of  b  as  4  to  3,  and  c  is  to  b  as  3  to  2.  Re- 
quired the  length  of  the  sides. 

Ans.  a  =  52.470,  £  =  39.353,  c  =  59.029. 
Prob.  12.  The  area  of  a  triangle  is  144,  the  base  is  24,  and 
one  of  the  angles  at  the  base  is  30°.     Required  the  other  sides 
of  the  triangle. 

Ans.  24  and  12.4233. 

Prob.  13.  Seven  men  bought  a  grinding-stone  of  60  inches 
diameter,  each  paying  one  seventh  part  of  the  expense.     What 
part  of  the  diameter  must  each  grind  down  for  his  share  ? 
Ans.  The  1st,  4.4508  inches ;  2d,  4.8400  inches ;  3d,  5.3535 
inches ;  4th,  6.0765  inches ;  5th,  '7.2079  inches ; 
6th,  9.3935  inches;  7th,  22.6778  inches. 
Prob.  14.  The  area  of  an  equilateral  triangle  is  17  square 
feet  and  83  square  inches.     What  is  the  length  of  each  side  ? 

Ans.  76.45  inches. 

Prob.  15.  The  parallel  sides  of  a  trapezoid  are  20  and  12  feet, 
and  the  other  sides  are  15  and  17  feet.  Required  the  area  of 
the  trapezoid. 

Ans.  240  square  feet. 


184  TRIGONOMETRY. 

Prob.  16.  How  many  square  yards  of -canvas  arc  required  to 
make  a  conical  tent  which  is  20  feet  in  diameter  and  12  feet 
high? 

Ans.  54.526  square  yards. 

Prob.  17.  The  circumference  of  an  hexagonal  pillar  is  7  feet, 
and  the  height  11  feet  2  inches.  Required  the  solid  contents  of 
the  pillar. 

Ans.  39.488  cubic  feet. 

Prob.  18.  The  base  of  the  great  pyramid  of  Egypt  is  a  square 
whose  side  measures  746  feet,  and  the  altitude  of  the  pyramid 
is  450  feet.  Required  the  volume  of  the  pyramid. 

Ans.  83,477,400  cubic  feet. 

Prob.  1 9.  A  side  of  the  base  of  a  frustum  of  a  square  pyra- 
mid is  25  inches,  a  side  of  the  top  is  9  inches,  and  the  height  is 
20  feet.  Required  the  volume  of  the  frustum. 

Ans.  43.102  cubic  feet. 

Prob.  20.  Three  persons,  having  bought  a  sugar-loaf,  would 
divide  it  equally  among  them  by  sections  parallel  to  the  base. 
It  is  required  to  find  the  altitude  of  each  person's  share,  suppos- 
ing the  loaf  to  be  a  cone  whose  height  is  20  inches. 

Ans.  13.8672,  3.6044,  and  2.5284  inches. 
Prob.  21.  If  a  cubical  foot  of  brass  were  to  be  drawn  into 
wire  of  one  thirtieth  of  an  inch  in  diameter,  it  is  required  to  de- 
termine the  length  of  the  said  wire,  allowing  no  loss  in  the  metal. 

Ans.  55003.94  yards ;  or  31  miles  443.94  yards. 
Prob.  22.  How  high  above  the  surface  of  the  earth  must  a 
person  be  raised  to  see  one  third  of  its  surface  ? 

Ans.  The  height  of  its  diameter. 

Prob.  23.  If  a  heavy  sphere,  whose  diameter  is  4  inches,  be 
let  fall  into  a  conical  glass  full  of  water,  whose  diameter  is  5, 
and  altitude  6  inches,  it  is  required  to  determine  how  much  wa- 
ter will  run  over. 

Ans.  26.272  cubic  inches. 

Prob.  24.  The  capacity  of  a  cylinder  is  a  cubic  feet,  and  its 
convex  surface  is  b  square  feet.  Required  the  dimensions  of 
the  cylinder. 

O  72 

Ans.  Radius  of  base  =  — -,  and  altitude =- — . 

b  4<2TT 

Prob.  25.  A  triangular  pyramid,  the  sides  of  whose  base  are 


EXAMPLES   FOR    PRACTICE.  185 

13,  14,  and  15  inches  respectively,  and  whose  altitude  is  16 
inches,  is  cut,  at  the  distance  of  2  inches  from  the  vertex,  by  a 
plane  parallel  to  the  base.  Required  the  volume  of  the  frustum 
of  the  pyramid. 

Ans.  447.125  cubic  inches. 

Prob.  26.  The  altitude  of  a  cone  is  10  inches,  and  the  radius 
of  its  base  is  5  inches.  At  what  distance  from  the  base  must  a 
plane  pass  parallel  to  the  base,  so  as  to  cut  off  a  frustum  whose 
capacity  is  20  cubic  inches  ? 

Ans.  0.2614  inches. 

SURVEYING. 

Prob.  1.  The  angle  of  elevation  of  a  spire  I  found  to  be  39° 
27' ',  and  going  directly  from  it  225  feet  on  a  horizontal  plane,  I 
found  the  angle  to  be  only  24°  38'.     "What  is  the  height  of  the 
spire,  and  the  distance  from  its  base  to  the  second  station  ? 
Ans.  Height  238.02  feet,  distance  508.18  feet. 

Prob.  2.  Wishing  to  know  the  distance  of  an  inaccessible  ob- 
ject, I  measured  a  horizontal  base-line  1328  feet,  and  found  the 
angles  at  the  ends  of  this  line  were  84°  23'  and  43°  19'.    What 
was  the  distance  of  the  object  from  each  end  of  the  base-line  ? 
Ans.  1151.44  feet,  and  1670.35  feet. 

Prob.  3.  Wishing  to  know  the  distance  between  two  inacces- 
sible objects,  C  and  D,  I  measured  a  base-line,  AB,  3784  feet, 
and  found  the  angle  BAD  =47°  32',  the  angle  DAC  =  39°  53', 
the  angle  ABC=46°  34',  and  the  angle  CBD-380  V.  What 
is  the  distance  from  C  to  D  ? 

Ans.  3257.36  feet. 

Prob.  4.  Suppose  a  light-house  built  on  the  top  of  a  rock ;  the 
distance  between  the  place  of  observation  and  that  part  of  the 
rock  which  is  level  with  the  eye,  and  directly  under  the  build- 
ing, is  1860  feet ;  the  distance  from  the  top  of  the  rock  to  the 
place  of  observation  is  2538  feet,  and  from  the  top  of  the  build- 
ing 2550  feet.  Required  the  height  of  the  light-house. 

Ans.  17  feet  7  inches. 

Prob.  5.  At  85  feet  distance  from  the  bottom  of  a  tower, 
standing  on  a  horizontal  plane,  the  angle  of  its  elevation  was 
found  to  be  52°  30'.  Required  the  altitude  of  the  tower. 

Ans.  IIOJ  feet. 


186  TRIGONOMETRY. 


6.  At  a  certain  station,  the  angle  of  elevation  of  an  in- 
accessible tower  was  26°  30"  ;  "but,  measuring  225  feet  in  a  di- 
rect line  toward  it,  the  angle  was  then  found  to  "be  51°  3(K 
Required  the  height  of  the  tower,  and  its  distance  from  the  last 
station.  Ans.  Height  186  feet,  distance  147  feet. 

Prob.  7.  To  find  the  distance  of  an  inaccessible  castle  gate,  I 
measured  a  line  of  73  yards,  and  at  each  end  of  it  took  the  an- 
gle of  position  of  the  object  and  the  other  end,  and  found  the  one 
to  be  90°,  and  the  other  61°  45'.  Required  the  distance  of  the 
castle  from  each  station. 

Ans.  135.8  yards,  and  154.2  yards. 

Prob.  8.  From  the  top  of  a  tower  143  feet  high,  by  the  sea- 
side, I  observed  that  the  angle  of  depression  of  a  boat  was  35°. 
"What  was  its  distance  from  the  bottom  of  the  tower  ? 

Ans.  204.22  feet. 

Prob.  9.  I  wanted  to  know  the  distance  between  two  places, 
A  and  B,  but  could  not  meet  with  any  station  from  whence  I 
could  see  both  objects.  I  measured  a  line  CD  =  200  yards  ;  from 
C  the  object  A  was  visible,  and  from  D  the  object  B  was  visi- 
ble, at  each  of  which  places  I  set  up  a  pole.  I  also  measured 
FC  —  200  yards,  and  DE  —  200  yards,  and  at  F  and  E  set  up 
poles.  I  then  measured  the  angle  AFC  =  83°,  ACF  =  54°  3V, 
ACD=53°  30',  BDC=:1560  25',  BDE  =  54°  30',  and  BED  = 
88°  30X.  Required  the  distance  from  A  to  B. 

Ans.  345.5  yards. 

Prob.  10.  From  the  top  of  a  light-house,  the  angle  of  depres- 
sion of  a  ship  at  anchor  was  3°  387,  and  at  the  bottom  of  the 
light-house  the  angle  of  depression  was  2°  43'.  Required  the 
horizontal  distance  of  the  vessel,  and  the  height  of  the  promon- 
tory above  the  level  of  the  sea,  the  light-house  being  85  feet 
high.  Ans.  Distance  5296.4  feet,  height  251.3  feet. 

Prob.  11.  An  observer,  seeing  a  cloud  in  the  west,  measured 
its  angle  of  elevation,  and  found  it  to  be  64°.  A  second  observ- 
er, situated  half  a  mile  due  east  from  the  first  station,  and  on 
the  same  horizontal  plane,  found  its  angle  of  elevation  at  the 
same  moment  of  time  to  be  only  35°.  Required  the  perpendic- 
ular height  of  the  cloud,  and  its  distance  from  each  observer. 

Ans.  Perpendicular  height  935.75  yards,  distances  1041.1 
and  1631.4  yards. 


EXAMPLES   FOR   PRACTICE.  187 

Prob.  12.  An  observer,  seeing  a  balloon  in  the  nort^i,  meas- 
ured its  angle  of  elevation,  and  found  it  to  be  36°  52'.  A  second 
observer,  situated  one  mile  due  south  from  the  first  station,  and 
on  the  same  horizontal  plane,  found  its  angle  of  elevation  at  the 
same  instant  to  be  only  30°  58'.  Required  the  perpendicular 
height  of  the  balloon,  and  its  distance  from  each  observer. 
Ans.  Perpendicular  height  3.003  miles,  distances  5.006 

and  5.837  miles. 

Prob.  13.  From  a  window  near  the  bottom  of  a  house  which 
seemed  to  be  on  a  level  with  the  bottom  of  a  steeple,  I  found  the 
angle  of  elevation  of  the  top  of  the  steeple  to  be  40°  ;  then  from 
another  window,  21  feet  directly  aboye  the  former,  the  like  angle 
was  37°  30'.  "What  was  the  height  and  distance  of  the  stee- 
ple ?  Ans.  Height  245.51  feet,  distance  292.59  feet. 

Prob.  14.  Wanting  to  know  my  distance  from  an  inaccessi- 
ble object,  P,  on  the  other  side  of  a  river,  and  having  no  instru- 
ment for  taking  angles,  but  a  chain  for  measuring  distances, 
from  each  of  two  stations,  A  and  B,  which  were  taken  at  300 
yards  asunder,  I  measured  in  a  direct  line  from  the  object  P  60 
yards,  viz.,  AC  and  BD  each  equal  to  60  yards ;  also,  the  diag- 
onal AD  measured  330  yards,  and  the  diagonal  BC  336  yards. 
What  was  the  distance  of  the  object  P  from  each  station  A 
and  B?  Ans.  AP  =  321.76  yards,  BP= 300.09  yards. 

Prob.  15.  Having  at  a  certain  (unknown)  distance  taken  the 
angle  of  elevation  of  a  steeple,  I  advanced  60  yards  nearer  on 
level  ground,  and  then  observed  the  angle  of  elevation  to  be  the 
complement  of  the  former.  Advancing  20  yards  still  nearer, 
the  angle  of  elevation  now  appeared  to  be  just  double  of  the 
first.  Required  the  altitude  of  the  steeple. 

Ans.  74.162  yards. 

Prob.  16.  In  a  garrison  there  are  three 
remarkable  objects,  A,  B,  C,  whose  dis- 
tances from  each  other  are  known  to  be, 
AB  213,  AC  424,  and  BC  262  yards.  I 
am  desirous  of  knowing  my  position  and 
distance  at  a  station,  P,  from  which  I  ob- 
served the  angle  APB,  13°  30',  and  the 
angle  CPB,  29°  50'. 
Ans.  AP  =  605.7122,  BP  =  429.6814,  CP  =  524.2365. 


188  TRIGONOMETRY. 

Prob.  17.  Supposing  the  object  B  to  be  on  the  opposite  side 
of  the  line  AC  (see  figure  to  Prob.  16),  and  that  the  distances  of 
the  objects  were,  AB  =  8  miles,  AC  =  12  miles,  and  BC  =  7|-  miles ; 
also,  the  angle  APB  =  19°,  and  the  angle  CPB=25°.  It  is  re- 
quired  to  find  the  distances  AP,  BP,  and  CP. 

Ans.  AP=9.4711  miles,  BP  =  16.3369  miles, 
CP  =  16.8485  miles. 

Prob.  18.  In  a  pentangular  field,  beginning  with  the  south 
side,  and  measuring  round  toward  the  east,  the  first  or  south 
side  was  27.35  chains,  the  second  31.15  chains,  the  third  23.70 
chains,  the  fourth  29.25  chains,  and  the  fifth  22.20  chains; 
also,  the  diagonal  from  the  first  angle  to  the  third  was  38.00 
chains,  and  that  from  the  third  to  the  fifth  was  40.10  chains. 
Required  the  area  of  the  field. 

Ans.  117  A.  2  72.  39  P. 

Prob.  19.  The  following  are  the  dimensions  of  a  five-sided 
field,  ABCDE:  the  side  AB  =  19.40  chains,  and  the  angle  B 
110°  30';  the  side  BC  =  15.55  chains,  and  the  angle  C  117° 
45' ;  the  side  CD =21.25  chains,  and  the  angle  D  91°  20' ;  and 
the  side  DE  =  27.41  chains.  Required  the  area  of  the  field. 

Ans.  66  A.  2  JR.  24  P. 

Prob.  20.  From  a  station,  H.  near  the  middle  of  a  field, 
ABCDEF,  from  which  I  could  see  all  the  angles,  I  measured 
the  distances  to  the  several  corners,  and  measured  the  angles 
formed  at  H  by  those  distances,  as  follows : 

Distances.  Angles. 

AH,  43.15  chains;  AHB,  60°  30'. 
BH,  29.82  «  BHC,  47  40 
CH,  35.61  "  CHD,  49  50 
DH,  50.10  "  DHE,57  10 
EH,  46.18  "  EHF,64  15 
FH,  36.06  "  FHA,  80  35 
Required  the  area  of  the  field. 

Ans.  412  A.  1  R.  17  P. 

NAVIGATION. 

Prob.  1.  From  a  ship  at  sea  I  observed  a  point  of  land  to 
bear  east  by  south,  and,  after  sailing  northeast  12  miles,  I  ol> 


EXAMPLES   FOR   PRACTICE.  189 

served  again,  and  found  its  bearing  to  be  southeast  ,by  east. 
How  far  was  the  last  observation  made  from  the  point  of  land  ? 

Ans.  26.07  miles. 

Prob.  2.  If  a  ship  in  latitude  50°  N.,  sails  52  miles  in  the  di- 
rection southwest  by  south,  what  latitude  has  she  arrived  in, 
and  how  much  farther  to  the  west  ? 

Ans.  Latitude  49°  16/8  N. ;  west,  28.9  miles. 

Prob.  3.  Two  ships  sail  from  the  same  port ;  the  one  sails 
east-northeast  85  miles,  the  other  sails  east  by  south  till  the  first 
ship  bears  northwest  by  west.  What  is  the  distance  of  the  sec- 
ond ship  from  the  port,  and  also  from  the  first  ship  ? 

Ans.  From  the  port,  184.7  miles ;   from  the  first  ship3 
123.4  miles. 

Prob.  4.  Two  ports  lie  east  and  west  of  each  other ;  a  ship 
sails  from  each,  namely,  the  ship  from  the  west  port  sails  north- 
east 89  leagues,  and  the  other  sails  80  leagues,  when  she  meets 
the  former.  Required  the  latter  ship's  course,  and  the  distance 
between  the  two  ports. 

Ans.  Course,  N.  38°  8/  W. ;  distance,  112.3  leagues. 

Prob.  5.  Two  ships  sail  from  a  certain  port;  the  one  sails 
south  by  east  45  leagues,  and  the  other  south-southwest  64 
leagues.  What  is  the  bearing  and  distance  of  the  first  ship  from 
the  second  ? 

Ans.  Bearing,  N.  65°  44'  E. ;  distance,  36.5  leagues. 

Prob.  6.  A  ship  sailing  northwest,  two  islands  appear  in 
sight,  of  which  the  one  bears  north,  and  the  other  west-north- 
west ;  but,  after  sailing  20  leagues,  the  former  bears  northeast, 
and  the  latter  west  by  south.  What  is  the  distance  asunder  of 
the  two  islands  ?  Ans.  32.38  leagues. 

Prob.  7.  To  a  vessel  sailing  on  a  certain  course,  a  headland 
was  observed  to  bear  due  west ;  four  hours  after  which  it  was 
seen  at  west-southwest ;  and  six  hours  after  this,  the  vessel  con- 
tinuing to  run  at  the  same  rate,  its  bearing  was  found  to  be 
south-southwest.  What  was  the  vessel's  course  at  the  time  ? 

Ans.  N.  42°  35'  W. 

Prob.  8.  Two  ships  of  war,  intending  to  cannonade  a  fort, 
are,  by  the  shallowness  of  the  water,  kept  so  far  from  it  that  they 
suspect  their  guns  can  not  reach  it  with  effect.  In  order,  there- 
fore, to  measure  the  distance,  they  separate  from  each  other  500 


190  TRIGONOMETRY. 

rods;  then  each  ship  observes  the  angle  which  the  other  ship 
and  the  fort  subtend,  which  angles  are  38°  16'  and  37°  9'. 
What,  then,  is  the  distance  between  each  ship  and  the  fort  ? 

Ans.  312  rods  and  320  rods. 

Prob.  9.  A  ship  from  the  latitude  42°  18'  N.,  sails  southwest 
by  south  until  her  latitude  is  40°  18'  N.  What  direct  distance 
has  she  sailed,  and  how  many  miles  has  she  sailed  to  the  west- 
ward? 

Ans.  Distance  run  144*3  miles,  and  has  sailed  to  west- 
ward 80.2  miles. 

Prob.  10.  A  ship  having  run  due  east  for  three  days,  at  the 
rate  of  eight  knots  an  hour,  finds  she  has  altered  her  longitude 
15  degrees.  What  parallel  of  latitude  did  she  sail  on  ? 

Ans.  Latitude  50°  12'. 

Prob.  11.  A  ship  in  latitude  43°  30'  N.,  and  longitude  44° 
W.,  sails  southeasterly  532  miles,  until  her  departure  from  the 
meridian  is  420  miles.  Required  the  course  steered,  and  the 
latitude  and  longitude  of  the  ship. 

Ans.  Course  S.  52°  8'  E.,  latitude  38°  3/5  N., 

longitude  34°  45'  W. 

Prob.  12.  A  ship  from  latitude  43°  20'  N.,  and  longitude  52° 
W.,  sails  E.S.E.  until  her  departure  is  745  miles.     Required 
the  distance  sailed,  and  the  latitude  and  longitude  of  the  ship. 
Ans.  Distance  808.4  miles,  latitude  38°  11/5  N., 

longitude  35°  36'  W. 

Prob.  13.  If  the  height  of  the  mountain  called  the  Peak  of 
Teneriffe  be  4  miles,  and  the  angle  -taken  at  the  top  of  it,  as 
formed  between  a  plumb-line  and  a  line  conceived  to  touch  the 
earth  in  the  horizon,  or  farthest  visible  point,  be  87°  25'  55",  it 
is  required  from  hence  to  determine  the  magnitude  of  the  whole 
earth,  and  the  utmost  distance  that  can  be  seen  on  its  surface 
from  the  top  of  the  mountain,  supposing  the  earth  to  be  a  per- 
fect sphere. 

Ans.  Distance  178.458  miles,  diameter  7957.793  miles. 
Prob.  14.  Required  the  course  and  distance  from  St.  Jago, 
one  of  the  Cape  Verd  islands,  in  latitude  14°  56'  N.,  to  the  island 
of  St.  Helena,  in  latitude  15°  45'  S.,  their  difference  of  longitude 
being  30°  12'. 

Ans.  Course  S.  44°  12'  E.,  distance  2567.8  miles, 


EXAMPLES   FOR   PRACTICE.  191 

Prob.  15.  A  ship  from  the  latitude  of  49°  57X  N.,  and  longi- 
tude of  30°  W.,  sails  S.  39°  W.,  till  she  arrives  in  the  latitude 
of  459  31X  N.  Required  the  distance  run,  and  the  longitude  of 
the  ship. 

Ans.  Distance  342.3  miles,  longitude  of  ship  35°  21X  W. 
Prob.  16.  Find  the  bearing  and  distance  from  San  Francisco, 
latitude  37°  48X  N.,  longitude  122°  28X  W.,  to  Jeddo,  latitude 
35°  40X  N.,  longitude  139°  40X  E.,  by  Mercator's  sailing. 

Ans.  Course  S.  88°  26X  "W.,  distance  4705  miles. 
Prob.  17.  Find  the  bearing  and  distance  from  San  Francisco 
to  Batavia  in  Java,  latitude  6°  9X  S.,  longitude  106°  53X  E.,  by 
Mercator's  sailing. 

Ans.  Course  S.  70°  12X  W.,  distance  7783  miles. 
Prob.  18.  Find  the  bearing  and  distance  from  San  Francisco 
to  Port  Jackson,  latitude  33°  51X  S.,  longitude  151°  14X  E.,  by 
Mercator's  sailing. 

Ans.  Course  S.  48°  18X  W.,  distance  6462  miles. 
Prob.  19.  Find  the  bearing  and  distance  from  San  Francisco 
to  Otaheite,  latitude  17°  29X  S.,  longitude  149°  29X  W.,  by  Mer- 
cator's sailing. 

Ans.  Course  S.  24°  44X  W.,  distance  3652  miles. 
Prob.  20.  Find  the  bearing  and  distance  from  San  Francisco 
to  Valparaiso,  latitude  33°  2X  S.,  longitude  71°  41X  W.,  by  Mer- 
cator's sailing. 

Ans.  Course  S.  33°  47X  E.,  distance  5354  miles. 

SPHERICAL  TRIGONOMETRY. 

Prob.  1.  Ill  the  right-angled  spherical  triangle  ABC,  there  are 
given  the  angle  C  23°  27X  42XX,  and  the  side  b  10°  39X  40/x. 
Required  the  angle  B,  and  the  sides  a  and  c. 

(a  =11°  35X49XX. 

Ans.  }  c  =  4°  35X  26X/. 

(B=66°58X    lxx. 

Prob.  2.  In  the  spherical  triangle  ABC,  the  side  BC  — 90°, 
the  side  AB  =  32°  57X  6/x,  and  the  side  AC  =  66°  32X.  Required 
the  angles. 

(A=132°    2X44XX. 

Ans.  <  B=  42°  56X  12XX. 

(  C=  23°  49X26//; 


192  TRIGONOMETRY. 

Prob.  3.  In  the  right-angled  spherical  triangle  ABO,  there 
are  given  the  angle  B  =  47°  54'  20",  and  the  angle  0  =  61°  50' 
29".  Required  the  sides. 

f  a  =  61°    4'  56". 

Ans.  \  b  =40°  30X20". 

(  c  =  50°  30'  30". 

Prob.  4.  In  the  spherical  triangle  ABC,  the  side  AC  =  90°, 
the  side  AB^115°  9',  and  the  angle  B  =  101°  40'.  Required 
the  remaining  parts. 

(  BC  =  113°  18'    7".' 

Ans.  JA    =  115°  54'  46". 

(C    =  117°  33'  49". 

Prob.  5.  In  the  spherical  triangle  ABC,  the  angle  A  =130° 
5X  22",  the  angle  C  =  36°  457  28",  and  the  side  AC  =44°  13X 
45X/-  Required  the  remaining  parts. 


BC  =  84°  14/29//. 
B  =  32°  26X    6". 

.  6.  In  the  spherical  triangle  ABC,  the  angle  A=33°  15' 
7",  B=31°  34X  38",  and  0  =  161°  25X  17".  Required  the  sides. 

(  a=  80°    5/    4". 

^.ws.  ]  b  =  70°  107  30". 

(c=145°    5X    2". 

ProZ>.  7.  In  the  spherical  triangle  ABC,  the  side  AB=rll2° 
22X  58",  AC  =  52°  39'  4",  and  BC^89°  16'  53".  Required 
the  angles. 

(  A=  70°  39X  0". 

Ans.  <  B'=  48°  36X  0". 

(€=119°  l^O". 

P/o&.  8.  In  the  spherical  triangle  ABC,  the  side  AB=r76°  So7 
36",  AC=:500  10X  30",  and  the  angle  A  =  34°  15'  3".  Re- 
quired  the  remaining  parts. 

(  B  =42°  15'  13". 
C  =121°  36'  20". 
BG=  40°  0X10". 

.  9.  The  latitudes  of  the  observatories  of  Paris  and  Pekin 
are  48°  50X  14"  N.  and  39°  54X  13"  N.,  and  their  difference  of 
longitude  is  114°  7/  30".  What  is  their  distance  ? 

Ans.  73°  56X  40". 


EXAMPLES   FOR   PRACTICE.  193 

Prob.  10.  Required  the  course  and  distance  from  New  York, 
latitude  40°  43'  N.,  longitude  74°  O7  W.,  to  San  Francisco,  lat- 
itude 37°  48X  N.,  longitude  122°  28'  W.,  on  the  shortest  route. 
Ana.  The  course  is  N.  78°  16X  "W. 

Distance,  2229.8  nautical  miles. 

Prob.  11.  Required  the  course  and  distance  from  San  Fran- 
cisco, latitude  37°  4SX  N.,  longitude  122°  28X  W.,  to  Jeddo,  in 
latitude  35°  40'  N.,  longitude  139°  40X  E.,  on  the  shortest 
route.  Ans.  The  course  is  N.  56°>41X  W. 

Distance,  4461.9  nautical  miles. 

Prob.  12.  Required  the  course  and  distance  from  San  Fran- 
cisco to  Batavia  in  Java,  latitude  6°  9X  S.,  longitude  106°  53X  E., 
on  the  shortest  route. 

Ans.  The  course  is  N.  67°  30X  W. 

Distance,  7516  nautical  miles. 

Prob.  13.  Required  the  course  and  distance  from  San  Fran- 
cisco to  Port  Jackson,  latitude  33°  51X  S.,  longitude  151°  14X  E., 
on  the  shortest  route. 

Ans.  The  course  is  S.  59°  50X  W. 

Distance,  6444  nautical  miles. 

Prob.  14.  Required  the  course  and  distance  from  San  Fran 
cisco  to  Otaheite,  latitude  17°  29X  S.,  longitude  149°  29X  W.,  on 
the  shortest  route. 

Ans.  The  course  is  S.  29°  45'  W. 

Distance,  3650.3  nautical  miles. 

Prdb.  15.  Required  the  course  and  distance  from  San  Fran^ 
cisco  to  Valparaiso,  latitude  33°  2X  S.,  longitude  71°  41  W.,  or 
the  shortest  route. 

Ans.  The  course  is  S.  39°  22X  E. 

Distance,  5108.5  nautical  miles. 

Prob.  16.  Suppose  two  ports,  one  in  north  latitude  30°,  and 
the  other  in  north  latitude  40°,  the  difference  of  longitude  be* 
tween  them  being  50°.     Required  the  bearing  and  distanco 
from  each  of  these*ports  to  an  island  that  lies  in  south  latitude 
18°,  and  which  is  equally  distant  from  both  of  the  said  ports 
Ans.  Bearing  from  first  port,      S.  40°  52X   9/x  E. 
Bearing  from  second  port,  S.  15    '9  47  W. 
The  distance,  59°  23X  19XX  =  3563.3  nautical  miles. 

THE    END. 

N 


TABLES 


or 


LOGARITHMS  OF  NUMBERS 


AND    OF 


SUES  AND  TANGENTS 


FOR   EVERY 


TEN  SECONDS  OF  THE  QUADRANT, 


WITH  OTHER  USEFUL  TABLES. 


BY   ELI  AS   LOO  MIS,  LL1X, 

PROFESSOR    OP  NATURAL  PHILOSOPHY   AND   ASTRONOMY   IN    YALE    COLLEGE,  AND    AUTHO3    OF 
"COURSE   OF   MATHEMATICS." 


TWENTY-FIFTH  EDITION. 


N  E  W  -  Y  O  R  K  : 

HARPER   &   BROTHERS,   PUBLISHERS, 
329    &    331    PEARL    STREET, 

FRANKLIN   SQUARE. 

186  8. 


Entered,  according  to  Act  of  Congress,  in  the  year  one  thousand  eight  hund  ed  and  forty-eight,  by 

HARPER  &  BROTHERS, 
in  the  Clerk's  Office  of  the  District  Court  of  the  Southern  District  of  New  York. 


CONTENTS. 


EXPLANATIC  H    i)F    THE    TABLES .  V 

TABLE    OF   LOGARITHMS   OF   NUMBERS 1 

LOGARITHMIC    SINES   AND    TANGENTS 21 

NATURAL    SINES   AND   TANGENTS ....  1  1 6 

NATURAL    SECANTS J  34 

LENGTHS   OF   CIRCULAR    ARCS 135 

TRAVERSE    TABLE 1  36 

ilERIDIONAL    PARTS •  H2 

CORRECTIONS   TO   MIDDLE   LATITUDE 1 49 

LOGARITHMS   FOR   COMPOUND   INTEREST,    ETC •          .      '  •  1 W 


PREFACE. 


THE  accompanying  tables  were  designed  to  afford  the  means  of  per- 
forming trigonometrical  computations  with  facility  and  precision.  The 
tables  chiefly  used  in  this  country  for  purposes  of  education  extend  to 
six  decimal  places,  like  those  in  the  present  collection;  but  the  pre- 
cision which  they  are  designed  to  furnish  is  only  attained  by  a  serious 
expenditure  of  labor.  In  the  Table  of  Logarithms  of  Numbers  they  do 
not  furnish  the  correction  for  a  fifth  figure  in  the  natural  number,  and 
the  labor  of  computing  this  correction  is  such  that  I  always  prefer  the 
use  of  Button's  Tables,  extending  to  seven  places,  even  in  computations 
to  which  six-place  logarithms  are  abundantly  competent.  In  the  pres 
ent  collection,  the  correction  for  a  fifth  figure  of  the  natural  number  is 
introduced  at  the  bottom  of  each  page,  and  the  table  is  thus  rendered 
nearly  as  useful  as  one  of  the  common  kind  extending  to  100,000.  The 
whole  has  been  carefully  compared  with  standard  authors,  and  nearly 
a  dozen  errors  have  thus  been  detected  in  the  common  tables. 

The  principal  table  in  this  collection  is  that  of  Logarithmic  Sines  and 
Tangents.  The  common  tables  in  this  country  extend  only  to  minutes, 
with  differences  to  100".  If,  in  a  trigonometrical  computation,  angles 
are  only  required  to  the  nearest  minute,  tables  to  five  places  are  quite 
sufficient ;  but  if  the  computation  is  to  be  carried  to  seconds,  these  can 
only  be  obtained  from  the  common  tables  by  a  great  expenditure  ol 
time  and  labor.  In  the  present  collection,  the  sines  and  tangents  are  fur- 
nished to  every  ten  seconds  of  the  quadrant,  and  at  the  bottom  of  each 
page  is  given  the  correction  for  any  number  of  seconds  less  than  ten,  so 
that  the  precision  of  seconds  can  be  obtained  with  almost  the  same  fa- 
cility as  that  of  minutes  with  the  tables  in  common  use.  Moreover, 
near  the  limits  of  the  quadrant,  by  means  of  an  auxiliary  table,  sines  and 
tangents  are  readily  obtained,  even  for  a  fraction  of  a  second.  The 
method  of  arrangement  of  the  sines  and  tangents  was  suggested  by  a 
table  in  M  ackay's  Longitude  ;  but  the  errors  of  that  table,  amounting  to 
several  thousand,  have  been  corrected  by  a  careful  comparison  with  the 
work  of  Ursinus.  By  comparison  with  the  same  standard,  more  than 
two  hundred  errors  (chiefly  in  the  final  figures)  have  been  detected  in 
the  tables  in  common  use. 

The  Table  of  Natural  Sines  and  Tangents  is  of  less  use  than  the  loga 
rithmic  ;  nevertheless,  it  is  often  important  for  reference,  particularly  in 
analytical  geometry  and  the  calculus  ;  and  it  is  useful  as  a  stepping- 
stone  to  assist  the  beginner  in  comprehending  the  nature  of  logarithmic 


IV 

sines  ar.d  tangents.  The  Traverse  Table  commonly  usea  in  fhis  country 
furnishes  the  latitude  and  departure  to  every  quarter  degree  of  the  quad 
rant,  for  distances  from  1  to  100,  and  occupies  ninety  pages.  The  accom- 
panying table  occupies  but  six  pages,  and  yields  ten  times  greater  pre- 
cision. 

The  Table  of  Meridional  Parts  extends  to  tenths  of  a  mile,  and  grea 
care  has  been  taken  to  insure  its  accuracy.     For  this  purpose,  I  have 
compared  all  the  similar  tables  within  my  reach,  and  among  them  have 
found  two  which  appeared  to  have  been  computed  independently.    Be 
tween  them  there  were  detected  674  discrepancies  in  the  final  figures. 
These  cases  were  all  recomputed,  and  78  errors  were  detected  in  the 
jest  copy  compared.     It  is  probable  that  the  numbers  in  this  table  are 
not  in  every  instance  true  to  the  nearest  tenth  of  a  mile  ;  but  it  is  be 
lieved  that  the  remaining  errors  are  few  in  number,  as  well  as  minute 
This  table  is  confidently  pronounced  more  accurate  than  any  similar 
one  with  which  I  have  been  able  to  compare  it. 

The  Table  of  Corrections  to  Middle  Latitude  was  computed  entirely 
anew.  The  corresponding  table  in  common  use,  which  was  originally 
computed  by  Workman,  contains  more  than  four  hundred  errors,  sev- 
eral of  them  amounting  to  two  minutes. 

On  the  whole,  it  is  believed  that  the  accompanying  tables  will  be 
found  more  convenient  to  the  computer  than  any  tables  of  six  decimal 
places  hitherto  published  in  this  country ;  and  that  they  will  be  pro- 
nounced sufficiently  extensive  for  all  purposes  of  academic  and  collegi- 
ate instruction,  as  well  as  for  practical  mechanics  and  surveyors, 


EXPLANATION  OF  THE  TABLES. 


TABLE  OF  LOGARITHMS  OF  NUMBERS,  pp.  1-20. 

LOGARITHMS  are  numbers  contrived  to  diminish  the  labor  of  Multiplica- 
tion and  Division  by  substituting  in  their  stead  Addition  and  Subtrac 
tion.  All  numbers  are  regarded  as  powers  of  some  one  number,  which 
is  called  the  base  of  the  system  ;  and  the  exponent  of  that  power  of  the 
base  which  is  equal  to  a  given  number,  is  called  the  logarithm  of  that 
number. 

The  base  of  the  common  system  of  logarithms  (called,  from  their  in- 
ventor, Briggs'  logarithms)  is  the  number  10.  Hence  all  numbers  •  are 
to  be  regarded  as  powers  of  10.  Thus,  since 

10°— 1,  0  is  the  logarithm  of  1  in  Briggs'  system  ; 

10'=  10,          1     "  "  10 

10'=  100,       2     "  «  100 

103=1000,      3     "  "  1000  "  " 

104=  10,000,  4     "  "  10,000  "  " 

&c.,  &c.,  &c. ; 

whence  it  appears  that,  in  Briggs'  system,  the  logarithm  of  every  num 
ber  between  1  and  10  is  some  number  between  0  and  1,  i.  e.,  is  a  prop- 
er fraction.  The  logarithm  of  every  number  between  10  and  100  is 
some  number  between  1  and  2,  i.  e.,  is  1  plus  a  fraction.  The  loga- 
rithm of  every  number  between  100  and  1000  is  some  number  between 
2  and  3,  i.  e.,  is  2  plus  a  fraction,  and  so  on. 

The  preceding  principles  may  be  extended  to  fractions  by  means  of 
negative  exponents.  Thus,  since 


10    =0.1,        — 1  is  the  logarithm  of  0.1  in  Briggs'  system 
10~z=0.01,       —2       "                 "           0.01  «  " 

0~3=0.001,     —3       "  "          0.001  "  " 

KT^O.OOOl,  —4      "  "          0.0001  "  " 

&c.,  &c.,  &c. 

Hence  it  appears  that  the  logarithm  of  every  number  between  1  and 

0.1  is  some  number  between  0  and  — 1,  or  may  be  represented  by 1 

plus  a  fraction ;  the  logarithm  of  every  number  between  0.1  and  .01  is 
some  number  between  — 1  and  —2,  or  may  be  represented  by — 2  plus 


vi  EXPLANATION   OF   THE   TABLES. 

a  fraction  ;  the  logarithm  of  every  number  between  .01  and  .001  is  seme 
number  between  — 2  and  — 3,  or  is  equal  to  — 3  plus  a  fraction,  and  so  on. 
The  logarithms  of  most  numbers,  therefore,  consist  of  an  integer  and 
a  'fraction.  The  integral  part  is  called  the  characteristic,  and  may  be 
known  from  the  following 

RULE. 

The  characteristic  of  the  logarithm  of  a  number  greater  than  unity -,  is 
OK<S  less  than  the  number  of  integral  figures  in  the  given  number. 

Thus  the  logarithm  of  297  is  2  plus  a  fraction  ;  that  is,  the  character- 
istic of  the  logarithm  of  297  is  2,  which  is  one  less  than  the  number  of 
integral  figures.  The  characteristic  of  the  logarithm  of  5673.29  is  3 ; 
that  of  73254.1  is  4,  &c. 

The  characteristic  of  the  logarithm  of  a  decimal  fraction  is  a  negative 
number,  and  is  equal  to  the  number  of  places  by  which  its  fast  significant 
figure  is  removed  from  the  place  of  units. 

Thus  the  logarithm  of  .0046  is  — 3  plus  a  fraction ;  that  is,  the  char- 
acteristic of  the  logarithm  is  — 3,  the  first  significant  figure  4  being 
removed  three  places  from  units. 

The  accompanying  table  contains  the  logarithms  of  all  numbers  from 
I  to  10,000  carried  to  6  decimal  places. 

To  find  the  Logarithm  of  any  Number  between  1  and  100. 

Look  on  the  first  page  of  the  table,  along  the  column  of  numbers  under 
N,  for  the  given  number,  and  against  it,  in  the  next  column,  will  be  found 
the  logarithm,  with  its  characteristic.  Thus, 

opposite  13  is  1.113943,  which  is  the  logarithm  of  13; 
65  is  1.812913,  "  «  65. 

To  find  the  Logarithm  of  any  Number  consisting  of  three  Figures. 

Look  on  one  of  the  pages  from  2  to  20,  along  the  left-hand  column 
narked  N,  for  the  given  number,  and  against  it,  in  the  column  headed  0, 
will  be  found  the  decimal  part  of  its  logarithm.     To  this  the  character 
stki  must  be  prefixed,  according  to  the  rule  already  given.     Thus 
the  logarithm  of  347  will  be  found,  from  page  8,  to  be  2.540329 ; 
«  "     871          "  "  "     18,    "     2.940018. 

As  the  first  two  figures  of  the  decimal  are  the  same  for  several  sue 
sessive  numbers  in  the  table,  they  are  not  repeated  for  each  logarithm 
icparately,  but  are  left  to  be  supplied.  Thus  the  decimal  part  of  the 
ogarithm  of  339  is  .530200.  The  first  two  figures  of  the  decimal  remain 
.he  same  up  to  347 ;  they  are  therefore  omitted  in  the 'table,  and  are  to 
oe  supplied. 

To  find  the  Logarithm  of  any  Number  consisting  of  four  Figures. 
Find  the  three  left-hand  figures  in  the  column  marked  N  as  before 


EXPLANATION  OP  THE  TABLES.          vii 

and  the  fourth  figure  at  the  head  of  one  of  the  other  columns.     Opposite 
to  tile  first  three  figures,  and  in  the  column  under  the  fourth  figure,  will 
be  found  four  figures  of  the  logarithm,  to  which  two  figures  from  the 
column  headed  0  are  to  be  prefixed,  as  in  the  former  case.     The  char- 
acteristic, must  be  supplied  by  the  usual  rule.     Thus 
the  logarithm  of  3456  is  3.538574 ; 
"  "     8765  is  3.942752. 

In  several  of  the  columns  headed  1,  2,  3,  &c.,  small  dots  are  lound  m 
the  place  of  figures.  This  is  to  show  that  the  two  figures  which  are  to 
be  prefixed  from  the  first  column  have  changed,  and  they  are  to  be 
taken  from  the  horizontal  line  directly  below.  The  place  of  the  dots  is 
to  be  supplied  with  ciphers.  Thus 

the  logarithm  of  2045  is  3.310693 ; 
"  "     9777  is  3.990206. 

The  two  leading  figures  from  the  column  0  must  also  be  taken  from 
the  horizontal  line  below,  if  any  dots  have  been  passed  over  on  the  same 
Horizontal  line.  Thus 

the  logarithm  of  1628  is  3.211654. 

To  find  the  Logarithm  of  any  Number  containing  more  than  four 

Figures. 

By  inspecting  the  table,  we  shall  find  that  within  certain  limits  the  log- 
arithms are  nearly  proportional  to  their  corresponding  numbers.     Thus 
the  logarithm  of  7250  is  3.860338  ; 
"     7251  is  3.860398  ; 
"  "     7252  is  3.860458  ; 

"  "     7253  is  3.860518. 

Here  the  difference  between  the  successive  logarithms,  called  the 
tabular  difference,  is  constantly  60,  corresponding  to  a  difference  of  unity 
in  the  natural  numbers.     If,  then,  we  suppose  the  logarithms  to  be  pro- 
portional to  their  corresponding  numbers  (as  they  are  nearly),  a  difl^r- 
ence  of  0.1  in  the  numbers  should  correspond  to  a  difference  of  6  it 
logarithms;  a  difference  of  0.2  in  the  numbers  should  correspond  to 
difference  of  12  in  the  logarithms,  &c.     Hence 

the  logarithm  of  7250.1  must  be  3.860344; 

"  "     7250.2         "       3.860350; 

"  «     7250.3         "       3.860356; 

&c.,  &c. 

In  order  to  facilitate  the  computation,  the  tabular  difference  is  insert- 
ed on  page  16  in  the  column  headed  D,  and  the  proportional  part  for  the 
fifth  figure  of  the  natural  number  is  given  at  the  bottom  of  the  page, 
Thus,  when  the  tabular  difference  is  60,  the  corrections  for  .1,  .2,  ,3, 
&c.,  are  seen  to  be  6,  12,  18,  &c. 

If  the  given  number  was  72501,  the  characteristic  of  its  logarithm 
"Should  be  4,  but  the  decimal  part  would  be  the  me  as  for  r/,5bO  J 


viii  EXPLANATION   OF   THE    TABLES. 

If  it  were  required  to  find  the  correction  for  a  sixth  figure  in  the  nut 
ural  number,  it  is  readily  obtained  from  the  Proportional  Parts  in  th^ 
table.     Thus,  if  the  correction  for  .5  is  30,  the  correction  for  .05  is  oh 
viously  3. 

As  the  differences  change  rapidly  in  me  first  part  of  the  table,  it 
was  found  inconvenient  to  give  the  proportional  parts  for  each  tabular 
difference ;  accordingly,  for  the  first  seven  pages  they  are  only  given 
for  the  even  differences,  but  the  proportional  parts  for  the  odd  differences 
will  be  readily  found  ^by  inspection. 
Required  the  logarithm  of  452789. 

The  logarithm  of  452700  is  5.655810 
The  tabular  difference  is  96. 

Accordingly,  the  correction  for  the  fifth  figure,  8,  is  77,  and  for  the 
sixth  figure,  9,  is  8.6,  or  9  nearly.  Adding  these  corrections  to  the  num- 
ber before  found,  we  obtain  5.655896. 

The  preceding  logarithms  do  not  pretend  to  be  perfectly  exact,  bu 
only  the  nearest  numbers  having  but  six  decimal  places.     Accordingly 
when  the  fraction  which  is  omitted  exceeds  half  a  unit  in  the  sixth  der; 
mal  place,  the  last  figure  must  be  increased  by  unity. 
Required  the  logarithm  of  8765432. 

The  logarithm  of  8765000  is  6.942752 

Correction  for  the  fifth  figure  4,  20 

"  "       sixth  figure  3,  1.5 

"  "       seventh  figure  2,  0.1 

Therefore  the  logarithm  of  8765432  is  6.942774. 

Required  the  logarithm  of  234567. 

The  logarithm  of  234500  is  5.370143 

Correction  for  the  fifth  figure  6,  111 

"              "      sixth  figure  7,  13 

Therefore  the  logarithm  of  234567  is  5.370267. 

To  find  the  Logarithm  of  a  Decimal  Fraction. 

The  decimal  part  of  the  logarithm  of  any  number  is  the  same  as  tha 
of  the  number  multiplied  or  divided  by  10,  100,  1000,  &c.  Hence,  for 
a  decimal  fraction,  we  find  the  logarithm  as  if  the  figures  were  integers 
and  prefix  the  characteristic  according  to  the  usual  rule. 

EXAMPLES. 

The  logarithm  of  345.6  is  2.538574 ; 

"        87.65  is  1.942752; 

"          2.345  is  0.370143; 

"  .1234  is  1.091315; 

»  "  005678  is  3.754195. 


EXPLANATION    OF    THE   TABLES.  ix 

The  minus  sign  is  placed  over  the  characteristic  to  show  that  thir 
alone  is  negative,  while  the  decimal  part  of  the  logarithm  is  positive. 

To  find  the  Logarithm  of  a  Vulgar  Fraction. 

We  may  reduce  the  vulgar  fraction  to  a  decimal,  and  find  its  log£ 
nthm  by  the  preceding  rule  ;  or,  since  the  value  of  a  fraction  is  equal  fc 
the  quotient  of  the  numerator  divided  by  the  denominator,  we  may  sub- 
tract the  logarithm  of  the  denominator  from  that  of  the  numerator  ;  the 
difference  will  be  the  logarithm  of  the  quotient. 
Required  the  logarithm  of  T\,  or  0.1875. 

From  the  logarithm  of  3,          0.477121,    . 
Subtract  the  logarithm  of  16,  1.204120. 
Leaves  logarithm  of  Ts¥,  or  .1875,  1.273001. 

Tn  the  same  manner  we  nnd 

the  logarithm  of  -fj  is  2.861697 ; 
"     jf£  is  1.147401. 

To  find  the  natural  Number  corresponding  to  any  Logarithm. 

Look  in  the  table  in  the  column  headed  0  for  the  first  two  figures  of 
the  logarithm,  neglecting  the  characteristic ;  the  other  four  figures  are 
to  be  looked  for  in  the  same  column,  or  in  one  of  the  nine  following  col- 
umns ;  and  if  they  are  exactly  found,  the  first  three  figures  of  the  corre- 
sponding number  will  be  found  opposite  to  them  in  the  column  headed 
N,  and  the  fourth  figure  will  be  found  at  the  top  of  the  page.  This  num 
ber  must  be  made  to  correspond  with  the  characteristic  of  the  given 
logarithm  by  pointing  off  decimals  or  annexing  ciphers.  Thus 

the  natural  number  belonging  to  the  logarithm  4.370143  is  23450  ; 

1.538574  is  34.56. 

If  the  decimal  part  of  the  logarithm  can  not  be  exactly  found  in  the 
table,  look  for  the  nearest  less  logarithm,  and  take  out  the  four  figures 
of  the  corresponding  natural  number  as  before ;  the  additional  figures 
may  be  obtained  by  means  of  the  Proportional  Parts  at  the  bottom  of 
the  page. 

Required  the  number  belonging  to  the  logarithm  4.368399. 

On  page  6,  we  find  the  next  less  logarithm  .368287. 

The  four  corresponding  figures  of  the  natural  number  are  2335. 
Their  logarithm  is  less  than  the  one  proposed  by  112.  The  tabular 
difference  is  186 ;  and,  by  referring  to  the  bottom  of  page  6,  we  find 
that,  with  a  difference  of  186,  the  figure  corresponding  to  the  Propor- 
tional Part  112  is  6.  Hence  the  five  figures  of  the  natural  number  are 
23356  ;  and,  since  the  characteristic  of  the  proposed  logarithm  is  4,  these 
five  figures  are  all  integral. 

Required  the  number  belonging  to  logarithm  5.345678. 

The  next  less  logarithm  in  the  table  is  .345570 

Their  difference  ''s  108. 


x  EXPLANATION   OF   THE   TABLES. 

The  first  four  figures  of  the  natural  number  are  2216. 
With  the  tabular  difference  196,  the  fifth  figure  corresponding  to  108 
is  seen  to  be  5,  with  a  remainder  of  10,  which  furnishes  a  sixth  figure  5 
nearly.     Hence  the  required  number  is  221655. 
In  the  same  manner  we  find 

the  number  corresponding  to  logarithm  3.538672  is  3456.78 ; 
«  "  "  1.994605  is  98.7654; 

«  "  "  T.647817  is  .444444. 

TABLE  OF  NATURAL  SINES  AND  TANGENTS,  pp.  116-133. 
This  is  a  table  of  natural  sines  and  tangents  for  every  degree  ana 
oninute  of  the  quadrant,  carried  to  six  places  of  figures.     Since  the  ra- 
dius of  the  circle  is  supposed  to  be  unity,  the  sine  of  every  arc  below 
90°  is  less  than  unity.     These  sines  are  expraesed  in  decimal  parts  of  the 
radius ;  and,  although  the  decimal  point  is  not  written  in  the  table,  it 
must  always  be  prefixed.     The  degrees  are  arranged  in  order  at  the  top 
of  the  page,  and  the  minutes  in  the  left  hand  vertical  column.     Directly 
under  the  given  number  of  degrees  at  the  top  of  the  page,  and  opposite 
to  the  minutes  on  the  left,  will  be  found  the  sine  required.     The  two 
leading  figures  are  repeated  at  intervals  of  ten  minutes.     Thus 
the  sine  of    6°  27'  is  .1 12336  ; 
"         "   28°  53'  is  .483028. 

The  same  number  in  the  table  is  both  the  sine  of  an  arc  and  the  co- 
sirie  of  its  complement.     The  degrees  for  the  cosines  must  be  sought  at 
the  bottom  of  the  page,  and  the  minutes  on  the  right.     Thus 
the  cosine  of  62°  25'  is  .463038 ; 
"  "  84°  23'  is  .097872. 

If  a  sine  is  required  for  an  arc  consisting  of  degrees,  minutes,  and  sec 
onds,  it  may  be  found  by  means  of  the  line  at  the  bottom  of  each  page, 
which  gives  the  proportional  part  corresponding  to  one  second  of  arc 
Required  the  sine  of  8°  9'  10". 

The  sine  of  8°  9' is  .141765. 

By  referring  to  the  bottom  of  page  116,  in  the  column  under  8°,  we 
find  the  correction  for  1"  is  4.80  ;  hence  the  correction  for  10"  must  be 
48,  which,  added  to  the  number  above  found,  gives  for  the  sine  of  8° 
9'  10",  .141813. 

In  the  same  manner  we  find 

the  cosine  of  56°  34'  28"  is  .550853. 

It  will  be  observed,  that  since  the  cosines  decrease  while  the  arcs  in 
Crease,  the  correction  for  the  28"  is  to  be  subtracted  from  the  cosine 
of  56°  34'. 

The  arrangement  of  the  table  of  natural  tangents  is  similar  to  that  of 
the  table  of  sines.  The  tangents  for  arcs  less  than  45°  are  all  less  than 
•-adius,  and  consist  wholly  of  decimals.  For  arcs  above  45°,  the  tan- 
gents are  all  greater  than  radius  and  contain  both  integral  and  decimal 


EXPLANATION   OF   THE    TABLES.  xi 

figures.     The  proportional  parts  at  the  bottom  of  each  page  enable  us 
readily  to  find  the  correction  for  seconds.     Thus 

the  natural  tangent  of  32°  29'  IS"  is  .636784  ; 
"  "      745  35'  55"  is  3.63014. 

To  find  the  Number  of  Degrees,  Minutes,  and  Seconds  belonging  to  a 
given  Sine  or  Tangent. 

If  the  given  sine  or  tangent  is  found  exactly  in  the  table,  the  corie- 
sponding  degrees  will  be  found  at  the  top  of  the  page,  and  the  minutes 
on  the  left  hand.  But  when  the  given  number  is  not  found  exactly  in 
the  table,  look  for  the  sine  or  tangent  which  is  next  less  than  the  pro- 
posed one,  and  take  out  the  corresponding  degrees  and  minutes.  Find, 
also,  the  difference  between  this  tabular  number  and  the  number  pro- 
posed, and  divide  it  by  the  proportional  part  for  V  found  at  the  bottom 
of  the  page ;  the  quotient  will  be  the  required  number  of  seconds. 

Required  the  arc  whose  sine  is  .750000. 

The  next  less  sine  in  the  table  is  .749919,  the  arc  corresponding  to 
which  is  48°  35'.  The  difference  between  this  sine  and  that  proposed 
is  81,  which,  divided  by  3.21,  gives  25.  Hence  the  required  arc  is  48° 
35'  25". 

In  the  same  manner  we  find 

the  arc  whose  tangent  is  2.000000,  to  be  63°  26'  6". 

TABLE  OF  NATURAL  SECANTS,  pp.  134—5. 

This  is  a  table  of  natural  secants  for  every  ten  minutes  of  the  quad- 
rant carried  to  seven  places  of  figures.  The  degrees  are  arranged  in 
order  in  the  first  vertical  column  on  the  left,  and  the  minutes  at  the  top 
of  the  page.  Thus 

the  secant  of  21°  20'  is  1.073561  ; 
"    81°  50'  is  7.039622. 

If  a  secant  is  required  for  a  number  of  minutes  not  given  in  the  table, 
the  correction  for  the  odd  minutes  may  be  found  by  means  of  the  last 
,<3rtical  column  on  the  right,  which  shows  the  proportional  part  for  one 
minute. 

Let  it  be  required  to  find  the  secant  of  30°  33'. 

The  secant  of  30°  30'  is  1.160592. 

The  correction  for  1'  is  198.9,  which,  multiplied  by  3,  gives  597 
Adding  this  to  the  number  before  found,  we  obtain  1.161189. 

For  a  cosecant,  the  degrees  must  be  sought  in  the  right-hand  vertical 
column,  and  the  minutes  at  the  bottom  of  the  page.     Thus 
the  cosecant  of  47°  40'  is  1.352742. 

TABLE  OF  LOGARITHMIC  SINES  AND  TANGENTS,  pp.  21-115. 

This  is  a  table  of  the  logarithms  of  the  sines  and  tangents  for  every 
ton  seconds  of  the  quadrant,  carried  to  six  places  of  decimals  The  de- 


xii  EXPLANATION    OF    THE    TABLES. 

grees  and  seconds  are  placed  at  the  top  of  the  page,  end  the  minutes  ,n 
the  left  vertical  column.  After  the  first  two  degrees,  the  three  leading 
figures  in  the  table  of  sines  are  only  given  in  the  column  headed  0",  and 
are  to  be  prefixed  to  the  numbers  in  the  other  columns,  as  in  the  table 
of  logarithms  of  numbers.  Also,  where  the  leading  figures  change,  this 
change  is  indicated  by  dots,  as  in  the  former  table.  The  correction  fo> 
any  number  of  seconds  less  than  10  is  given  at  the  bottom  of  the  page. 

To  find  the  Logarithmic  Sine,  or  Tangent  of  a  given  Arc. 

Look  for  the  degrees  at  the  top  of  the  page,  the  minutes  on  the  lelt 
hand,  and  the  next  less  tenth  second  at  the  top  ;  then,  under  the  seconds, 
and  opposite  to  the  minutes,  will  be  found  four  figures,  to  which  the 
three  leading  figures  are  to  be  prefixed  from  the  column  headed  0" ;  to 
this  add  the  proportional  part  for  the  odd  seconds  from  the  bottom  of 
the  page. 

Required  the  logarithmic  sue  of  24°  27'  34". 

The  logarithmic  sine  of  24°  27'  30"  is  9.617033. 
Proportional  part  for  4"  is  18. 

Logarithmic  sine  of  24°  27'  34"  is       9.617051. 

This  is  the  logarithm  of  .414049  found  in  the  table  of  natural  sines  on 
page  120.  The  natural  sine  being  less  than  unity,  the  characteristic 
of  its  logarithm  is  negative.  Tc  ooviate  this  inconvenience,  the  char- 
acteristics in  the  table  nave  all  been  increased  by  10 ;  or  the  logarith- 
mic sines  may  be  regarded  as  the  logarithms  of  natural  sines  computed 
for  a  radius  of  10,000,000,000. 

Required  the  logarithmic  tangent  of  73°  35'  43". 

The  logarithmic  tangent  of  73°  35'  40"  is  10.531031. 
Proportional  part  for  3"  is  23.    1 

Logarithmic  tangent  of  73°  35'  43"  10.531054. 

When  a  cosine  is  required,  the  degrees  and  seconds  must  be  sought 
at  the  bottom  of  the  page,  and  the  minutes  on  the  right,  and  the  correc 
tion  for  the  odd  seconds  must  be  subtracted  from  the  number  in  the  table 
Required  the  logarithmic  cosine  of  59°  33'  47". 

The  logarithmic  cosine  of  59°  33'  40"  is  9.704682. 
Proportional  part  for  7"  is  25. 

Logarithmic  cosine  of  59°  33'  47"  is  9.704657. 
So,  also,  the  logarithmic  cotangent  of  37°  27'  14"  is  found  to  be  10.115744, 
The  proportional  parts  given  at  the  bottom  of  each  page  correspond 
to  the  degrees  at  the  top  ol  the  page  increased  by  30',  and  are  not 
strictly  applicable  to  any  other  number  of  minutes ;  nevertheless,  the 
differences  of  the  sines  change  so  slowly,  except  near  the  commence- 
ment of  the  quadrant,  that  the  error  resulting  from  using  these  numbers 
for  every  part  of  the  page  will  sa  iorn  exceed  a  unit  in  the  sixth  deci- 
mal place.  For  the  first  two  degrees,  the  differences  change  so  rapidly 


EXPLANATION    OF   THE    TABLES.  x i i i 

the  prDportional  part  for  1"  is  given  for  each  minute  in  the  right- 
hand  column  of  the  page.  The  correction  for  any  number  of  seconds 
less  than  ten  will  be  found  by  multiplying  the  proportional  part  for  i' 
by  the  given  number  of  seconds. 

Required  the  logarithmic  sine  of  1°  17'  33". 

The  logarithmic  sine  of  1°  11'  30"  is  8.352991 

The  correction  for  3"  is  found  by  multiplying  93.4  by  3,  which  gives 
280.     Adding  this  to  the  above  tabular  number,  we  obtain 
the  sine  of  1°  11'  33",  8.353271. 

A  similar  method  may  be  employed  for  several  of  the  first  degrees 
of  the  quadrant,  if  the  proportional  parts  at  the  bottom  of  the  page  are 
not  thought  sufficiently  precise.  This  correction  may,  however,  be  ob- 
tained pretty  nearly  by  inspect'on  from  comparing  the  proportional 
parts  for  two  successive  degrees.  Thus,  on.  page  26,  the  correction  for 
1",  corresponding  to  the  sine  of  2°  30',  is  48  ;  the  correction  for  1",  cor- 
responding to  the  sine  of  3°  30',  is  34.  Hence  the  correction  for  1", 
corresponding  to  the  sine  of  3°  0',  must  be  about  41  ;  and  in  the  same 
manner  we  may  proceed  for  any  other  part  of  the  table. 

Near  the  close  of  the  quadrant,  the  tangents  vary  so  rapidly,  that  the 
same  arrangement  of  the  table  is  adopted  as  for  the  commencement  of 
the  quadrant.  For  the  last  as  well  as  the  first  two  degrees  of  the  quad- 
rant, the  proportional  part  to  1"  is  given  for  each  minute  separately. 
These  proportional  parts  are  computed  for  the  minutes  placed  opposite 
to  them,  increased  by  30',  and  are  not  strictly  applicable  to  any  other 
number  of  seconds  ;  nevertheless,  the  differences  for  the  most  part 
change  so  slowly,  that  the  error  resulting  from  using  these  numbers  for 
every  part  of  the  same  horizontal  line  is  quite  small.  When  great  ac- 
curacy is  required,  the  table  on  page  114  may  be  employed  for  arcs 
near  the  limits  of  the  quadrant.  This  table  furnishes  the  differences  be- 
tween the  logarithmic  sines  and  the  logarithms  of  the  arcs  expressed  in 
seconds.  Thus 

the  logarithmic  sine  of  0°  5'  is  7.162696; 
the  logarithm  of  300"  (=5')  is  2.477121 ; 

the  difference  is  4.685575. 

This  is  the  number  found  on  page  114,  under  the  heading  log.  SITU 
A — log.  A",  opposite  5  min. ;  and  in  a  similar  manner  the  other  numbers 
in  the  same  column  are  obtained.  These  numbers  vary  quite  slowly 
for  two  degrees  ;  and  hence,  to  find  the  logarithmic  sine  of  an  arc  less 
than  two  degrees,  we  have  but  to  add  the  logarithm  of  the  arc  expressed 
iii  seconds  to  the  appropriate  number  found  in  this  table. 
Required  the  logarithmic  sine  of  0°  7'  22". 

Tabular  number  from  page  114,  4.685575. 
The  logarithm  of  442"  is  2.645422. 

Logarithmic  sine  of  0°  7'  22"  is  7.330997 


xiv  EXPLANATION   OF   THE    TABLES. 

The  logarithmic  tangent  of  an  arc  less  than  two  degrees  is  found  la 
a  similar  manner. 
Required  the  logarithmic  tangent  of  0°  21'  36". 

Tabular  number  from  page  114,       4.685584. 

The  logarithm  of  1656"  is  3.219060. 

Logarithmic  tangent  of  0°  27'  36"  is  7.904644. 

The  column  headed  log.  cot.  A+log.  A"  is  found  by  adding  the  log- 
arithmic cotangent  to  the  logarithm  of  the  arc  expressed  in  seconds. 
Hence,  to  find  the  logarithmic  cotangent  of  an  arc  less  than  two  degrees, 
we  must  subtract  from  the  tabular  number  the  logarithm  of  the  arc  in 
seconds. 

Required  the  logarithmic  cotangent  of  0°  27'  36". 

Tabular  number  from  page  114,  15.314416. 

The  logarithm  of  1656"  is  3.219060. 

Logarithmic  cotangent  of  0°  27'  36"  is  12.095356. 
The  same  method  will,  of  course,  furnish  cosines  and  cotangents  oj 
arcs  near  90°. 

The  secants  and  cosecants  are  omitted  in  this  table,  since  they  are 
easily  derived  from  the  cosines  and  sines. 

The  logarithmic  secant  is  found  by  subtracting  the  logarithmic  cosine 
from  20  ;  and  the  logarithmic  cosecant  is  found  by  subtracting  the  loga- 
rithmic sine  from  20. 

Thus  we  have  found  the  logarithmic  sine  of  24°  27'  34"  to  be  9.617051. 
Hence  the  logarithmic  cosecant  of  24°  27'  34"  is  10.382949. 

The  logarithmic  cosine  of  54°  12'  40"  is  9.767008. 

Hence  the  logarithmic  secant  of  54°  12'  40"  is  10.232992. 

To  find  the  Arc  corresponding  to  a  given  Logarithmic  Sine  or  Tangent 

If  the  given  number  is  found  exactly  in  the  table  the  corresponding 
degrees  and  seconds  will  be  found  at  the  top  of  the  page,  and  the  min- 
utes on  the  left.  But  when  the  given  number  is  not  found  exactly  in 
the  table,  look  for  the  sine  or  tangent  which  is  next  less  than  the  pro- 
posed one,  and  take  out  the  corresponding  degrees,  minutes,  and  sec- 
onds. Find,  also,  the  difference  between  this  tabular  number  and  the 
number  proposed,  and,  corresponding  to  this  difference,  at  the  bottom  of 
the  page  will  be  found  a  certain  number  of  seconds,  which  is  to  be  added 
.c  the  arc  before  found. 

Required  the  arc  corresponding  to  the  logarithmic  sine  9.750000. 

The  next  less  sine  in  the  table  is  9.749987. 

The  arc  corresponding  to  which  is  34°  13'  0". 

The  difference  between  its  sine  and  the  one  proposed  is  13,  corre- 
sponding to  which,  at  the  bottom  of  the  page,  we  find  4"  nearly.  Hence 
ihf*  required  arc  is  34°  13'  4-". 


EXPLANATION  OF  THE  TABLES.          xv 

In  the  same  manner  we  find  tne  arc  corresponding  to  logarithmic  tan 
gent  10.250000,  to  be  60°  38'  57". 

When  the  arc  falls  within  the  first  two  degrees  of  the  quadrant,  the 
odd  seconds  may  be  found  by  dividing  the  difference  between  the  tab- 
ular number  and  the  one  proposed,  by  the  proportional  part  for  1".  We 
thus  find  the  arc  corresponding  to  logarithmic  sine  8.400000,  to  be  1° 
26'  22"  nearly. 

We  may  employ  the  same  method  for  the  last  two  degrees  of  the 
quadrant  when  a  tangent  is  given ;  but  near  the  limits  of  the  quadrant 
it  is  better  to  employ  the  auxiliary  table  on  page  114.  If  we  subtract 
the  corresponding  tabular  number  on  page  114  from  the  given  logarith 
mic  sine,  the  remainder  will  be  the  logarithm  of  the  arc  expressed  in 
seconds. 

Required  the  arc  corresponding  to  logarithmic  sine  7.000000. 

We  see,  from  page  22,  that  the  arc  must  be  nearly  3' ;  the  correspond 
ing  tabular  number  on  page  114  is  4.685575. 

The  difference  is  2.314425; 
which  is  the  logarithm  of  206."265. 

Hence  the  required  arc  is  3'  26."265. 

In  the  same  manner  we  find  the  arc  corresponding  to  logarithmic 
tangent  8.184008,  to  be  0°  52'  35". 

TABLE  FOR  THE  LENGTHS  OF  CIRCULAR  ARCS,  p.  135. 

This  table  contains  the  lengths  of  every  single  degree  up  to  60,  and 
at  intervals  often  degrees  up  to  180  ;  also  for  every  minute  and  second 
up  to  20.  The  lengths  are  all  expressed  in  decimal  parts  of  radius. 

Required  the  length  of  an  arc  of  57°  17'  44."8. 

Take  out  from  their  respective  columns  the  lengths  answering  to  each 
of  these  numbers  singly,  and  add  them  all  together  thus : 

57°     .     .    \     .     .     .     0.9948377 

17' 0049451 

40" 0001939 

4" .0000194 

0."8  .  .0000039 


The  sum  is        1.0000000. 

That  is,  the  length  of  an  arc  of  57°  17'  44."8  is  equal  to  the  radius  of 
the  circle. 

TRAVERSE  TABLE,  pp.  136-141. 

This  table  shows  the  difference  of  latitude  and  the  departure  to  four 
decimal  places  for  distances  from  1  to  10,  and  for  bearings  from  0°  to 
90°,  at  intervals  of  15'.     If  the  bearing  is  less  than  45°,  the  angle  will 
oe  found  on  the  left  margin  of  one  of  the  pages  of  the  table,  and  the  dis 
tance  at  the  top  or  bottom  of  the  page  ;  the  difference  of  latitude  wil 


xvi          EXPLANATION  OF  THE  TABLES 

be  found  in  the  column  headed  lat.  at  the  top  of  the  page,  and  the  ae> 
parture  in  the  column  headed  dep.  If  the  bearing  is  more  than  45°y  ine 
angle  will  be  found  on  the  right  margin,  and  the  difference  of  atitude 
tvill  be  found  in  the  column  marked  lat.  at  the  bottom  of  the  page,  and 
the  departure  in  the  other  column.  The  latitudes  and  departures  for 
different  distances  with  the  same  bearing,  are  proportional  to  the  dis- 
tances. Therefore  the  distances  may  be  reckoned  as  tens,  hundreds,  01 
thousands,  if  the  place  of  the  decimal  point  in  each  departure  and  differ- 
ence of  latitude  be  changed  accordingly. 

Required  the  latitude  and  departure  for  the  distance  32.25,  and  the 
bearing  10°  30'. 

On  page  136,  opposite  to  10°  30',  we  find  the  following  latitudes  and 
departures,  proper  attention  being  paid  to  the  position  of  the  decimal 
7oints. 

Distance.  Diff.  Lat.  Dep. 

30  29.498  5.467 

2  1.966  .364 

.2  .197  .036 

.05  .049  .009 


32.2.5  31.710  5.876. 

TABLE  OF  MERIDIONAL  PARTS,  pp.  142—148. 

This  table  gives  the  length  of  the  enlarged  meridian  on  Mercators 
Chart  to  every  minute  of  latitude  expressed  in  geographical  miles  and 
tenths  of  a  mile.  The  degrees  of  latitude  are  arranged  -in  order  at  the 
top  of  the  page,  and  the  minutes  on  both  the  right  and  left  margins. 
Under  the  degrees  and  opposite  to  the  minutes  are  placed  the  merid- 
ional parts  corresponding  to  any  latitude  less  than  80°.  Thus 
the  meridional  parts  for  latitude  12°  23'  are  748.9; 

57°  42'  are  4260.5. 

TABLE  OF  CORRECTIONS  TO  MIDDLE  LATITUDE,  p.  149. 

This  table  is  used  in  Navigation  for  correcting  the  middle  latitude 
The  given  middle  latitude  is  to  be  found  either  in  the  first  or  last  verti- 
cal column,  opposite  to  which,  and  under  the  given  difference  of  latitude, 
is  inserted  the  proper  correction  in  minutes,  to  be  added  to  the  middle 
latitude  to  obtain  the  latitude  in  which  the  meridian  distance  is  accu 
rately  equal  to  the  departure.  Thus,  if  the  middle  latitude  is  41°,  and 
the  difference  of  latitude  14°,  the  correction  will  be  found  to  be  25', 
which,  added  to  the  middle  latitude,  gives  the  corrected  middle  latitude 
41°  25'. 


...  .  ^ 

A  TABLE 

i 

OJS 

LOGARITHMS  OF  NUMBERS 

FROM    1    TO    10,000. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

i 

2 

3 
4 
5 

o.oooooo 
o.3oio3o 
0.477121 
0.602060 
0.698970 

26 
27 
28 
29 
3o 

.414973 
.43x364 
.447i58 
.462398 
.477121 

5i 

52 

53 

54 
55 

.707570 
.716003 
.724276 
.732394 
•74o363 

76 

77 
78 

79 

80 

.880814 
.886491 

.892095 

.897627 

.qoSogo 

6 

8 

.9 
10 

0.778151 
0.845098 
0.903090 
0.954243 
i  .000000 

3i 

32 

33 

34 
35 

.491362 
.5o5!5o 
.5i85i4 
.53x479 
.544o68 

56 

57 

58 

«9 
60 

.748188 
.755875 
.763428 
.770852 
.778151 

81 
82 
83 
84 
85 

.  9o8485 
•9i38i4 
.919078 
.924279 
.929419 

ii 

12 

i3 

i4 
i5 

.041393 
.079181 
.113943 
.146128 
.176091 

36 

37 
38 
39 
4o 

.5563o3 

.568202 
.579784 
.591065 
.602060 

61 
62 
63 
64 
65 

.78533o 
.792392 
.799341 
.806180 
.812913 

86 
87 
88 

89 
90 

.934498 
.9395i9 

.944483 
.949390 
.954243 

16 

I7 

18 

19 

20 

.2o4l20 

.2^0449 
.255273 
.278754 
.3oio3o 

4i 
4a 
43 
44 
45 

.612784 
.623249 
.633468 
.643453 
.653213 

66 
67 
68 
69 
7° 

.819544 
.826075 
.832509' 
.838849 
.845098 

91 
92 
93 

9i 
95 

.959041 
.963788 
.968483 
.978128 
.977724 

21 
22 
23 
24 
25 

.322219 

.342423 
.361728 
i.38o2ii 
1.397940 

46 

47 
48 

49 
5o 

.662758 
.672098 
.681241 
.690196 
.698970 

7i 

72 

73 

74 
75 

.85i258 
.857332 
.863323 
.869232 
.875061 

96 

97 
98 

99 

IOO 

.982271 
.986772 
.991226 
.995635 

2.OOOOOO 

N.B.  In  the  following  table,  the  two  leading  figures  in  the  first  column 
of  logarithms  are  to  be  prefixed  to  all  the  numbers  of  the  same  horizontal 
line  in  the  next  nine  columns  ;  but  when  a  point  (.)  occurs,  its  place  is  to 
bs  supplied  by  a  cipher,  and  the  two  leading  figures  are  to  be  taken  from 
the  next  lower  line. 

LOGARITHMS    OF    NUMBERS. 


N 

0     1 

2 

3 

4 

5 

6 

7 

8 

9   D. 

100 

o  loooo 

o434 

0868 

i3oi 

1734 

2166 

2598 

3o29 

346  1 

389i 

432 

101 

4321 

475i 

5i8i 

5609 

6o38 

6466 

6894 

7321 

7748 

8174 

428 

IO2 

8600 

0,026 

945i 

9876 

.3oo 

.724 

ii47 

i57o 

i993 

24i5 

424 

io3 

OI2837 

3259 

368o 

4ioo 

452i 

536o 

5779 

6197 

6616 

419 

104 

7o33 

745i 

7868 

8284 

87oo 

9116 

9532 

9947 

.36i 

.775 

4i6 

io5 

021189 

i6o3 

2016 

2428 

2841 

3252 

3664 

4o75 

4486 

412 

106 

53o6 

57i5 

6i25 

6533 

6942 

735o 

7757 

8i64 

857i 

8978 

4o8 

107 

9384 

9789 

.I95 

.600 

ioo4 

i4o8 

1812 

2216 

2619 

3021 

4o4 

108 

o33424 

3826 

4227 

4628 

5029 

543o 

583o 

623o 

6629 

7028 

4oo 

109 

•7426 

7825 

8223 

8620 

9OI7 

94i4 

98ii 

.207 

.602 

.998 

396 

no 

o4i393 

1787 

2182 

2576 

2969 

3362 

3755 

4i48 

454o 

4932 

393 

in 

5323 

5?i4 

6io5 

6495 

6885 

7275 

7664 

8o53 

8442 

883o 

389 

112 

9218 

9606 

9993 

.38o 

.766 

ii53 

i538 

1924 

2309 

2694 

386 

ii3 

o53o78 

3463 

3846 

423o 

46i3 

4996 

5378 

5760 

6142 

6524 

382 

ii4 

69o5 

7286 

7666 

8o46 

8426 

88o5 

9i85 

9563 

9942 

.320 

3?9 

ii5 

060698 

io?5 

i452 

1829 

2206 

2582 

2958 

3333 

3709 

4o83 

376 

116 

4458 

4832 

5206 

558o 

5953 

6326 

6699 

7071 

7443 

78i5 

373 

117 

8186 

8557 

8928 

9298 

9668 

..38 

.407 

.776 

ii45 

i5i4 

369 

118 

o7i882 

225o 

2617 

2985 

3352 

37i8 

4o85 

445  1 

4816 

5i82 

366 

119 

554? 

5912 

6276 

664o 

7oo4 

7368 

773i 

8o94 

8457 

88i9 

363 

120 

9181 

9543 

9904 

.266 

.626 

.987 

1  347 

1707 

2067 

2426 

36o 

121 

o82785 

3i44 

35o3 

386i 

4219 

4576 

4934 

5291 

5647 

6oo4 

357 

-N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1). 

434 

43 

87 

i3o 

1  74 

217 

260 

3o4 

347 

391 

432 

43 

86 

i  So 

i73 

216 

259 

302 

346 

389 

43o 

43 

86 

129 

172 

2l5 

258 

3oi 

344 

387 

428 

43 

86 

128 

2l4 

257 

3oo 

342 

385 

426 

43 

85 

128 

170 

213 

256 

298 

34i 

383 

424 

42 

85 

I27 

170 

212 

254 

297 

339 

382 

422 

42 

84 

I27 

169 

211 

253 

295 

338 

38o 

420 

42 

84 

126 

168 

SIO 

262 

294 

336 

378 

4x8 

42 

84 

125 

167 

2O9. 

25l 

293 

334 

376 

| 

4i6 

42 

83 

125 

166 

208 

250 

291 

333 

3  74 

4i4 

4i 

83 

124 

166 

207 

248 

290 

33i 

373 

• 

4l2 

4i 

82 

124 

i65 

206 

247 

288 

33o 

37i 

4io 

4i 

82 

123 

1  64 

2O5 

246 

287 

328 

369 

4o8 

4i 

82 

122 

1  63 

204 

245 

286 

326 

367 

4o6 

4i 

81 

122 

162 

203 

244 

284 

325 

365 

4o4  & 

4o 

81 

121 

lC2 

2O2 

242 

283 

333 

364 

402  £ 

4o 

80 

121 

161 

201 

241 

281 

322 

362 

£ 

4oo  Q* 

4o 

80 

120 

1  60 

200 

24O 

280 

320 

36o 

§ 

398  § 

4o 

80 

II9 

i59 

199 

239 

279 

3i8 

358 

£  < 
& 

396  .o< 

4o 

79 

i58 

198 

238 

277 

3i7 

356 

394  ! 

39 

79 

1x8 

168 

197 

236 

276 

3i5 

355 

_ 

392  g 

39 

78 

118 

i57 

I96 

235 

274 

3i4 

353 

39o  &H 

39 

78 

117 

1  56 

i95 

234 

273 

3l2 

35i 

388 

39 

78 

116 

i55 

i94 

233 

272 

3io 

349 

386 

39 

77 

116 

1  54 

i93 

232 

270 

3o9 

347 

384 

38 

77 

ii5* 

1  54 

I92 

230 

269 

3o7 

346 

382 

38 

76 

ii5 

i53 

229 

267 

3o6 

344 

38o 

38 

76 

u4 

l52 

I90 

228 

266 

3o4 

342 

378 

38 

76 

ii3 

i5i 

i89 

227 

265 

302 

34o 

376 

38 

75 

ii3 

i5o 

1  88 

226 

263 

3oi 

338 

374 

37 

75 

112 

i5o 

187 

224 

262 

299 

337 

372 

37 

74 

112 

149 

186 

223 

260 

298 

335 

37o 

37 

74 

III 

i48 

i85 

222 

269 

296 

333 

368 

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203 

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230 

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1  68 

106 

224 

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278 

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167 

195 

222 

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276 

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266  |  < 

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LOGARITHMS    OF    NUMBERS. 


N.    0 

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6 

7 

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5542 

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230 

189 

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7354 

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225 

194 

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121 

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162 

182 

200 

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60 

80 

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180 

198 

20 

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59 

79 

99 

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139 

168 

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LOGARITHMS    OF    LUMBERS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

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231 

344392 

4589 

4785 

4981 

5i78 

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196 

222 

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195 

223 

83o5 

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8889 

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9278 

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9860 

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194 

224 

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1023 

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193 

225 

2i83 

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2568 

2761 

2954 

3i47 

3339 

3532 

3724 

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226 

4io8 

43oi 

4493 

4685 

4876 

5o68 

5260 

5452 

5643 

5834 

192 

227 
228 

6026 
7935 

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6599 
85o6 

6790 
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6981 

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191 

190 

229 

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4176 

4363 

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5ii3 

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232 

5488 

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6049 

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6423 

6610 

6796 

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187 

233 

7356 

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180 

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260 

4973 

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166 

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263 

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0 

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1  88 

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LOGARITHMS    OF    NUMBERS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

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264 

421604 

1768 

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2097 

2261 

2426 

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266 

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3574 

3737 

3901 

4o65 

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4392 

4555 

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266 

4882 

5o45 

5208 

537i 

5534 

5697 

586o 

6o23 

6186 

6349 

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267 

65xi 

6674 

6836 

6999 

7161 

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7973 

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8783 

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1042 

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270 

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2167 

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28o9 

271 

2969 

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36io 

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4090 

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272 

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283 

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284 

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292 

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6126 

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657i 

67i9 

293 

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294 

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LOGARITHMS    OF    J\  u  M  B  E  u  s. 


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6664 

6785 

69o5 

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7i46 

7267 

7387 

120 

36i 

75o7 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

SSSg 

N. 

0 

1 

2 

3 

4   ||   5 

6 

7 

8 

9 

D. 

i38 

i4 

28 

4i 

55 

69 

83 

97 

no 

124 

i36  A 

i4 

27 

4i 

54 

68 

82 

95 

109 

122 

1  34  £ 

i3 

27 

4o 

54 

67 

80 

94 

I07 

121 

| 

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i3 

26 

4o 

53 

66 

79 

92 

1  06 

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i3o  "3 

CJ  J 

1  3 

26 

39 

52 

65 

78 

91 

104 

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128  £ 

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26 

38 

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64 

77 

90 

102 

ii5 

P. 

126  o 

i3 

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38 

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63 

76 

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ii3 

124  §• 

12 

25 

37 

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62 

74 

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12 

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61 

73 

85 

98 

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12 

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72 

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1  08 

LOGARITHMS    OF    NUMBERS. 


!  N. 

0 

1 

2 

3 

4 

5 

6 

7    8 

9 

D. 

j  362 

5587o9 

8829 

8948 

9o68 

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.743 

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364 

56noi 

1221 

1  34o 

i459 

i578 

i698 

1817 

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2o55 

2I74 

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365 

2293 

24l2 

253i 

265o 

2769 

2887 

3oo6 

3i25 

3244 

3362 

366 

348l 

36oo 

37i8 

3837 

3955 

4074 

4l92 

43ii 

4429 

4548 

5<57 

4666 

4784 

49o3 

5021 

5i39 

5257 

5376 

5494 

56i2 

573o 

118 

368 

5848 

5966 

6o84 

6202 

6320 

6437 

6555 

6673 

6791 

69o9 

369 

7026 

7i44 

-7262 

7379 

7497 

76i4 

7732 

7849 

7967 

8084 

37o 

8202 

83i9 

8436 

8554 

8671 

8788 

89o5 

9O23 

9140 

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9374 

949i 

96o8 

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372 

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0660 

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IOIO 

1126 

1243 

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1476 

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373 

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2o58 

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229I 

2407 

2523 

2639 

2755 

116 

374 

2872 

2988 

3io4 

322O 

3336 

3452 

3568 

3684 

38oo 

39i5 

375 

4o3i 

4i47 

4263 

4379 

4494 

46  1  o 

4726 

484i 

4957 

5o72 

376 

5i88 

53o3 

54i9 

5534 

565o 

5765 

588o 

5996 

6m 

6226 

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377 

634i 

6457 

6572 

6687 

6802 

69i7 

7032 

7147 

7262 

7377 

378 

7492 

76o7 

7722 

7S36 

795i 

8066 

8181 

8295 

84io 

8525 

379 

8639 

8754 

8868 

8983 

9°97 

O2I2 

9326 

944i 

9555 

9669 

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38o 

9784 

9898 

.  .12 

.126 

.241 

.355 

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.583 

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.811 

38i 

58o925 

io39 

ii53 

I267 

i38i 

i495 

1608 

1722 

i835 

i95o 

382 

2o63 

2I77 

229I 

2404 

25i8 

263i 

2745 

2858 

2972 

3o85 

383 

3i99 

33i2 

3426 

3539 

3652 

3765 

3879 

3992 

4io5 

4218 

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1  384 

433i 

4444 

4557 

467o 

4783 

4896 

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5l22 

5235 

5348 

i  385 

546  1 

5574 

5686 

5799 

5912 

6024 

6i37 

625o 

6362 

6475 

386 

6587 

6700 

6812 

6925 

7o37 

7i49 

7262 

7374 

7486 

7599 

112 

387 

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7935 

8o47 

8160 

8272 

8384 

8496 

8608 

8-720 

388 

8832 

8944 

9o56 

9l67 

9279 

939i 

95o3 

96i5 

9.726 

9838 

389 

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.953 

390 

59io65 

11-76 

128-7 

i399 

i5io 

1621 

1732 

i843 

1955 

2066 

III 

391 

2177 

2288 

2399 

25lO 

2621 

2732 

2843 

2954 

3o64 

3i75 

392 

3286 

3397 

35o8 

36i8 

3-729 

384o 

SgSo 

4o6i 

4171 

4282 

393 

4393 

45o3 

46i4 

4724 

4834 

4945 

5o55 

5i65 

5276 

5386 

no 

394 

5496 

56o6 

6717 

5827 

5937 

6o47 

6i57 

6267 

6377 

6487 

395 

6597 

67o7 

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6927 

7o37 

7i46 

7256 

7366 

7586 

396 

7695 

78o5 

79i4 

8024 

8i34 

8243 

8353 

8462 

8572 

8681 

397 

879i 

8900 

9oo9 

9119 

0,228 

9337 

9446 

9556 

9665 

9774 

109 

398 

9883 

9992 

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.537 

.646 

.755 

.864 

399 

6oo973 

1082 

II9I 

1299 

i4o8 

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1734 

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4oo 

2060 

2l69 

2277 

2386 

2494 

26o3 

2711 

28i9 

2928 

3o36 

108 

4oi 

3i44 

3253 

336i 

3469 

3577 

3686 

3794 

39O2 

4oio 

4n8 

402 

4226 

4334 

444a 

455o 

4658 

4766 

4874 

4982 

5089 

5i97 

4o3 

53o5 

54i3 

552i 

5628 

5736 

5844 

595i 

6o59 

6166 

6274 

\  4o4 

C33i 

6489 

6596 

67o4 

6811 

69i9 

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7i33 

724l 

7348 

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4o5 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8206 

83i2 

84i9 

4o6 

8526 

8633 

874o 

8847 

8954 

9o6i 

916-7 

9274 

938i 

9488 

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98o8 

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4o8 

610660 

o767 

o873 

0979 

1086 

II92 

1298 

i4o5 

i5ii 

161-7 

1  06 

N. 

0 

1 

2 

3 

4  ||  5 

6 

7 

8 

9 

D. 

n9 

12 

24 

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48 

60 

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83 

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107 

118 

12 

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35 

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59 

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12 

23 

35 

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116  g 

12 

23 

35 

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58 

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81 

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ii5  £ 

12 

23 

35 

46 

58 

69 

81 

92 

104 

ii4  ~ 

II 

23 

34 

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57 

68 

80 

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II 

23 

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45 

57 

68 

79 

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102 

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112  | 

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22 

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67 

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II 

22 

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89 

100 

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22 

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54 

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76 

86 

97 

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21 

32 

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54 

64 

75 

86 

96 

10 


LOGARITHMS    OF    NUMBEUS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

409 

611723 

1829 

1936 

2042 

2l48 

2254 

236o 

2466 

2572 

2678 

1  06 

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2784 

2890 

2996 

3l02 

3207 

33i3 

3419 

3525 

363o 

3736 

4n 

3842 

3947 

4o53 

4i59 

4264 

4370 

4475 

458i 

4686 

4792 

4l2 

4897 

5oo3 

5io8 

52i3 

53i9 

5424 

5529 

5634 

574o 

5845 

io5 

4i3 

595o 

6o55 

6160 

6265 

6370 

6476 

658i 

6686 

6790 

6895 

4i4 

7000 

yioS 

7210 

73i5 

7420 

7525 

7629 

7734 

7839 

7943 

4i5 

8o48 

8i53 

8257 

8362 

8466 

857I 

8676 

878o 

8884 

8989 

416 

9o93 

9198 

9302 

94o6 

95n 

96!5 

9719 

9824 

9928 

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104 

4i7 

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o656 

0760 

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1072 

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1176 

1280 

1  384 

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1592 

i695 

1799 

I9o3 

2007 

2IIO 

419 

22l4 

23i8 

2421 

2525 

2628 

273,2 

2835 

2939 

3o42 

3i46 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4i79 

io3 

421 

4282 

4385 

4488 

459i 

4695 

4798 

49oi 

5oo4 

5107 

0210 

422 

53i2 

54i5 

55i8 

562i 

5724 

5827 

5929 

6o32 

6i35 

6238 

428 

634o 

6443 

6548 

6648 

675i 

6853 

6956 

7o58 

7161; 

7263 

424 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8i85 

8287 

IO2 

425 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

93o8 

426 

9410 

9512 

9613 

97i5 

9817 

9919 

.  .21 

.123 

.224 

.326 

427 

63o428 

o53o 

o63i 

o733 

o835 

o936 

io38 

nSg 

i24r 

1  342 

428 

1  444 

1  545 

1  647 

I748 

1849 

ig5i 

2O52 

2i53 

2256 

2356 

101 

429 

2457 

255g 

2660 

276l 

2862 

2963 

3o64 

3i65 

3266 

3367 

43o 

3468 

3569 

367o 

377i 

3872 

3973 

4074 

4i75 

4276 

4376 

43  1 

4477 

4578 

4679 

4779 

488o 

498i 

5o8i 

5i82 

5283 

5383 

IOO 

432 

5484 

5584 

5685 

5785 

5886 

5986 

6087 

6i87 

6287 

6388 

433 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

720.0 

739o 

434 

7490 

759o 

7690 

779° 

7890 

799° 

8090 

8190 

829o 

8389 

435 

8489 

858g 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

99  ' 

436 

9486 

9586 

9686 

9785 

9885 

9984 

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.i83 

.283 

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437 

64o48i 

o58i 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

i375 

438 

i474 

i573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3o58 

3i56 

3255 

3354 

44o 

3453 

355i 

365o 

3749 

3847 

3g46 

4o44 

4i43 

4242 

434o 

98 

44i 

4439 

4537 

4636 

4734 

4832 

493i 

5029 

5127 

5226 

5324 

442 

5422 

552i 

5619 

57i7 

58i5 

SgiS 

6011 

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6208 

63o6 

443 

64o4 

65o2 

6600 

6698 

670.6 

6894 

6992 

7089 

7^7 

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444 

7383 

748i 

7579 

7676 

7774 

7872 

7969 

8067 

8i65 

8262 

445 

836o 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9i4o 

9237 

97 

446 

9335 

9432 

953o 

9627 

9724 

9821 

99i9 

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447 

65o3o8 

o4o5 

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o599 

0696 

o793 

o89o 

0987 

io84 

1181 

448 

1278 

i375 

l472 

i569 

1666 

1762 

i859 

1956 

2o53 

2i5o 

449 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3oi9 

3n6 

45o 

32i3 

3309 

34o5 

35o2 

3598 

3695 

379i 

3888 

3984 

4o8o 

96 

45  1 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

485o 

4946 

5o42 

452 

5i38 

5235 

533i 

5427 

5523 

5619 

57i5 

58io 

59o6 

6002 

453 

6098 

6194 

629o 

6386 

6482 

6577 

6673 

6769 

6864 

696o 

454 

7o56 

7l52 

7247 

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7438 

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7629 

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8011 

8107 

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8298 

8393 

8488 

8584 

8679 

8774 

8870 

95 

456 

8965 

9060 

9I55 

9260 

9346 

944i 

9536 

963i 

9726 

9821 

457 

9916 

.  .  II 

.106 

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.296 

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.681 

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N. 

0 

1 

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5 

6 

7 

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D. 

1  06 

II 

21 

32 

42 

53 

64 

74 

85 

96 

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II 

21 

32 

42 

53 

63 

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84 

95 

104  g 

10 

21 

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42 

52 

62 

73 

83 

94 

w 

103  £ 

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21 

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4i 

52 

62 

72 

82 

93 

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61 

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101   §  ' 

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61 

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60 

70 

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29 

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77 

36 

LOGARITHMS    OF    J\UMBERS. 


N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

458 

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0960 

io55 

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1623 

1718 

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2OO2 

2096 

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238o 

2475 

2569 

2663 

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2758 

2852 

2947 

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3i35 

323o 

3324 

34i8 

35i2 

36o7 

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46i 

3yoi 

3795 

3889 

3983 

4o78 

4172 

4266 

436o 

4454 

4548 

462 

4642 

4736 

483o 

4924 

5oi8 

5lI2 

5206 

5299 

5393 

5487 

463 

558i 

5675 

5769 

5862 

5956 

6o5o 

6i43 

6237 

633i 

6424 

464 

65i8 

6612 

67o5 

6799 

6892 

6986 

7079 

7i73 

7266 

736o 

465 

7453 

7546 

764o 

7733 

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7920 

8oi3 

8106 

8i99 

8293 

& 

466 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9o38 

9i3i 

9224 

467 

93i7 

94io 

95o3 

9596 

9689 

9782 

9875 

9967 

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468 

670246 

o339 

o43i 

o524 

0617 

O7IO 

0802 

o895 

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1080 

469 

n73 

1265 

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i45i 

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i636 

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1821 

I9i3 

2OO5 

4yo 

2098 

2I9O 

2283 

2375 

2467 

256o 

2652 

2744 

2836 

2929 

92 

47i 

3021 

3n3 

32o5 

3297 

3390 

3482 

3574 

3666 

3758 

385o 

472 

3942 

4o34 

4126 

4218 

43io 

4402 

4494 

4586 

4677 

4-769 

473 

486i 

4953 

5o45 

5i37 

5228 

5320 

54i2 

55o3 

5595 

5687 

474 

5778 

587o 

5962 

6o53 

6:45 

6236 

6328 

64i9 

65n 

6602 

475 

6694 

6785 

6876 

6968 

7o59 

7i5i 

7242 

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7424 

75i6 

91 

476 

76o7 

7698 

7789 

7881 

7972 

8o63 

8i54 

8245 

8336 

842  7 

477 

85i8 

8609 

8700 

879i 

8882 

8973 

9o64 

9i55 

9246 

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478 

9428 

9519 

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97oo 

9791 

9882 

9973 

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479 

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o97o 

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1422 

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48  1 

2i45 

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2326 

2416 

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2596 

2686 

2777 

2867 

2957 

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3o47 

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3227 

33i7 

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3497 

3587 

3677 

3767 

3857 

483 

3947 

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4127 

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43o7 

4396 

4486 

4576 

4666 

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484 

4845 

4935 

5o25 

5u4 

52o4 

5294 

5383 

5473 

5563 

5652 

485 

5742 

583i 

592I 

6010 

6100 

6i89 

6279 

6368 

6458 

6547 

89 

486 

6636 

6726 

68i5 

69o4 

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7o83 

7I72 

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735i 

744o 

487 

7529 

7618 

7707 

7796 

7886 

7975 

8o64 

8:53 

8242 

833i 

488 

8420 

85o9 

8598 

8687 

8776 

8865 

8953 

9042 

9i3i 

922O 

489 

93o9 

9398 

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9575 

9664 

9753 

984i 

993o 

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490 

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0-728 

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3639 

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3727 

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4342 

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45i7 

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46o5 

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4868 

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6o94 

6182 

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87 

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6356 

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6618 

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6793 

6880 

6968 

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73i7 

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7578 

7665 

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7839 

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8oi4 

499 

8101 

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8275 

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9067 

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9578 

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5oi 

9838 

9924 

.  .11 

..08 

.184 

.271 

.358 

.444 

.53i 

.617 

502 

700704 

o79o 

o877 

0963 

io5o 

n36 

1222 

i3o9 

i395 

i48a   36 

5o3 

i568 

i654 

i74i 

1827 

I9i3 

i999 

2086 

2I72 

2258 

2344 

5o4 

243l 

25l7 

26o3 

2689 

2775 

2861 

2947 

3o33 

3n9 

32o5 

5o5 

3291 

3377 

3463 

3549 

3635 

372I 

38o7 

3893 

3979 

4o65 

5o6 

4i5i 

4236 

4322 

44o8 

4494 

4579 

4665 

475i 

4837 

4922 

5o7 

5oo8 

5o94 

5i79 

5265 

535o 

5436 

5522 

56o7 

5693 

5778 

N. 

0 

1 

2 

3 

4  ||  5 

6 

7 

8 

9 

D. 

95 

10 

19 

29 

38 

48 

57 

67 

76 

86 

94  » 

9 

J9 

28 

38 

47 

56 

66 

75 

85 

93  S 

9 

i9 

28 

37 

47 

56 

65 

74 

84 

<e 

92  Pu 

9 

18 

28 

37 

46 

55 

64 

74 

83 

c 

9i  1 

9 

18 

27 

36 

46 

55 

64 

73 

82 

1' 

90  1   9 

18 

27 

36 

45 

54 

63 

72 

81 

89  S    9 

18 

27 

36 

45 

53 

62 

71 

80 

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88  2i   9 

18 

26 

35 

44 

53 

62 

70 

79 

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17 

26 

35 

44 

52 

61 

70 

78 

86   I  9 

17 

26 

34 

43 

5a 

60 

69 

77 

i  2 


LOGARITHMS    OF    NUMBERS. 


1  —  —  

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

5o8 

705864 

5g49 

6o35 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

65 

609 

6718 

68o3 

6888 

6974 

7o59 

7i44 

7229 

73i5 

74oo 

7485 

5io 

757° 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

825i 

8336 

5n 

8421 

85o6 

859i 

8676 

8761 

8846 

893i 

9oi5 

9100 

9i85 

5l2 

9270 

9355 

944o 

9524 

9609 

9694 

9779 

9863 

9948 

..33 

5i3 

7x0117 

O202 

0287 

o37i 

o456 

o54o 

o625 

0710 

0794 

o879 

5x4 

o963 

io48 

Il32 

I2I7 

i3oi 

i385 

1470 

i554 

1639 

I723 

84 

5i5 

1807 

1892 

1976 

2060 

2i44 

2229 

23i3 

2397 

2481 

2566 

616 

265o 

2734 

2818 

2902 

2986 

3o7o 

3i54 

3238 

3323 

34o7 

5x7 

349i 

3575 

3659 

3742 

3826 

3910 

3994 

4o78 

4162 

4246 

5x8 

433o 

44i4 

4497 

458i 

4665 

4749 

4833 

49i6 

5ooo 

5o84 

619 

5i67 

525i 

53-35 

54i8 

55o2 

5586 

5669 

5753 

5836 

592O 

620 

6oo3 

6087 

6170 

6254 

6337 

6421 

65o4 

6588 

667i 

6754 

83  j 

621 

6838 

6921 

7oo4 

7o88 

7171 

7254 

7338 

742I 

75o4 

7587 

622 

7671 

7754 

7837 

7920 

8oo3 

8086 

8i69 

8253 

8336 

84i9 

628 

85o2 

8585 

8668 

875i 

8834 

89i7 

9ooo 

9o83 

9i65 

9248 

624 

933i 

94i4 

9^97 

958o 

9663 

9745 

9828 

99u 

9994 

••77 

626 

720159 

0242 

o325 

o4o7 

0490 

o573 

o655 

o738 

0821 

o9o3 

626 

0986 

1068 

n5i 

1233 

i3i6 

i398 

i48i 

i563 

1  646 

1-728 

82 

627 

1811 

i893 

i975 

2o58 

2l4o 

2222 

23o5 

2387 

2469 

2552 

628 

2634 

2716 

2798 

2881 

2963 

3o45 

3127 

3209 

3291 

3374 

629 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4o3o 

4lI2 

4i94 

53o 

4276 

4358 

444o 

4522 

46o4 

4685 

4767 

4849 

493I- 

5oi3 

53i 

5o95 

5i76 

5258 

534o 

5422 

55c3 

5585 

5667 

5  748 

583o 

532 

5912 

5993 

6075 

6i56 

6238 

6320 

64oi 

6483 

6564 

6646 

533 

6727 

6809 

6890 

6972 

7o53 

7i34 

7216 

7297 

7379 

746o 

81 

534 

754i 

7623 

77°4 

7785 

7866 

7948 

8o29 

8110 

8191 

8273 

535 

8354 

8435 

85i6 

8597 

8678 

8759 

884i 

8922 

gooS 

9o84 

536 

9i65 

9246 

9327 

94o8 

9489 

957° 

965i 

9732 

9813 

9893 

537 

9974 

..55 

.136 

.2I7 

.298 

.378 

•  459 

.54o 

.621 

.702 

538 

730782 

o863 

0944 

1024 

no5 

1186 

1266 

1  347 

1428 

i5o8 

539 

i589 

1669 

1750 

i83o 

1911 

1991 

2072 

2l52 

2233 

a3i3 

54o 

2394 

2474 

2555 

2635 

2715 

2796 

2876 

2956 

3o37 

3n7 

80 

.  54i 

3i97 

3278 

3358 

3438 

35i8 

3598 

8679 

3759 

3839 

39i9 

542 

3999 

4079 

4i6o 

4240 

4320 

44oo 

448o 

456o 

464o 

4720 

543 

48oo 

488o 

496o 

5o4o 

5l20 

6200 

5279 

5359 

5439 

55i9 

544 

5599 

5679 

5759 

5838 

5918 

5998 

6078 

6i57 

6237 

63i7 

545 

6397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7o34 

7ix3 

546 

7193 

7272 

7352 

743i 

75n 

759o 

767o 

7749 

7829 

7908 

79 

54? 

7987 

8067 

8i46 

8225 

83o5 

8384 

8463 

8543 

8622 

8701 

548 

8781 

8860 

8939 

9018 

9°97 

9i77 

9256 

9335 

94i4 

9493 

549 

9572 

965i 

973i 

9810 

9889 

9968 

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55o 

74o363 

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0521 

0600 

o678 

o757 

o836 

o9i5 

o994 

io73 

55i 

Il52 

1230 

1  309 

1  388 

i467 

i546 

1624 

I7o3 

I782 

1860 

552 

1939 

2018 

2096 

2I75 

2254 

2332 

2411 

2489 

2568 

2647 

553 

2725 

2804 

2882 

2961 

3o39 

3n8 

3196 

3275 

3353 

343i 

78 

554 

35io 

3588 

3667 

3745 

3823 

3902 

398o 

4o58 

4i36 

42  1  5 

555 

4293 

437i 

4449 

4528 

46o6 

4684 

4-762 

484o 

49i9 

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556 

5o75 

5i53 

523i 

53o9 

5387 

5465 

5543 

562i 

5699 

5777 

557 

5855 

5933 

6011 

6089 

6i67 

6245 

6323 

64oi 

6479 

6556 

558 

6634 

6712 

6790 

6868 

6945 

7023 

7IOI 

7179 

7266- 

7334 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D.  • 

86 

9 

-7 

26 

34 

43 

52 

60 

69 

77 

85  9 

9 

17 

26 

34 

43 

5i 

60 

68 

77 

w 

84  * 

8 

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25 

34 

42 

5o 

59 

67 

76 

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83  * 

8 

17 

25 

33 

42 

5o 

58 

66 

75 

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82   |. 

8 

16 

25 

33 

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49 

57 

66 

74 

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81  | 

8 

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24 

32 

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49 

57 

65 

73 

P 

80  | 

8 

16 

24 

32 

4o 

48 

56 

64 

72 

79  £ 

8 

16 

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32 

4o 

47 

55 

63 

7: 

78  * 

8 

16 

23 

3i 

39 

47 

55 

62 

70 

JU  O  G  A  K  I  T  II  M  S      OF      JN  U  M  B  E  R  d. 


N. 

0 

I 

2 

3 

4 

5 

6    7 

8 

9  [  D. 

559 

747412 

7489 

7567 

7645 

7722 

78oo 

7878 

7955 

8o33 

8uo|  78 

56o 

8188 

8266 

8343 

8421 

8498 

8576 

8653 

873i 

8808 

8885  1  i" 

56i 

8963 

9o4o 

9n8 

9i95 

9272 

935o 

9427 

95o4 

9582 

9659 

662 

9736 

98i4 

989i 

9968 

..45 

.123 

.200 

.277 

.354 

.43 

563 

75o5o8 

o586 

o663 

o74o 

0817 

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o97i 

io48 

1125 

I2O2 

564 

I279 

i356 

i433 

i5io 

i587 

i664 

i74i 

1818 

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1972 

565 

2048 

2125 

22O2 

2279 

2356 

2433 

25o9 

2586 

2663 

274O 

566 

2816 

2893 

2970 

3o47 

3i23 

320O 

3277 

3353 

343o 

35o6 

567 

3583 

366o 

3736 

38i3 

3889 

3966 

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4272 

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568 

4348 

44s  5 

45oi 

4578 

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473o 

48o7 

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4$  60 

5o36 

76 

56o 

5lI2 

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5265 

534i 

54i7 

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6180 

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6332 

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6864 

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7016 

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7927 

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573 

8:55 

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8533 

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8685 

876i 

8836 

574 

89ia 

8988 

9o63 

9i39 

92l4 

9290 

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575 

9668 

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75 

576 

76o422 

o498 

o573 

o649 

0724 

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o95o 

IO25 

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i326 

1402 

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i552 

1627 

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578 

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2378 

2453 

2529 

2604 

579 

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2754 

2829 

2904 

2978 

3o53 

3  1  28 

32o3 

3278 

3353 

58o 

3428 

35o3 

3578 

3653 

3727 

38o2 

3877 

3952 

4027 

4ioi 

58i 

4i76 

425i 

4326 

44oo 

4475 

455o 

4624 

4699 

47?4 

4848 

582 

4923 

4998 

5o72 

5i47 

5221 

5296 

537o 

5445 

5520 

5594 

583 

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5743 

58i8 

589a 

5966 

6o4i 

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6i9o 

6264 

6338 

74 

584 

64i3 

6487 

6562 

6636 

67io 

6785 

6859 

6933 

7oo7 

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585 

7i56 

723o 

73o4 

7379 

7453 

7527 

7601 

7675 

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586 

7898 

7972 

8o46 

8120 

8i94 

8268 

8342 

84i6 

849o 

8564 

587 

8638 

87I2 

8786 

8860 

8934 

9oo8 

9o82 

9i56 

923o 

93o3 

588 

9377 

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982O 

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589 

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o336 

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o7o5 

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io73 

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I22O 

I293 

1367 

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1661 

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1808 

1881 

i955 

2028 

2102 

2I75 

2248 

73 

592 

2322 

2395 

2468 

2542 

26i5 

2688 

2762 

2835 

29o8 

2081 

593 

3o55 

3i28 

3201 

3274 

3348 

3421 

3494 

3567 

364o 

3713 

594 

3786 

386o 

3933 

4oo6 

4o79 

4i52 

4225 

4298 

437i 

4444 

595 

45i7 

459o 

4663 

4736 

48o9 

4882 

4955 

5028 

5  1  oo 

5i73 

596 

5246 

53i9 

5392 

5465 

5538 

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5683 

5756 

5829 

59O2 

597 

5974 

6o47 

6120 

6i93 

6265 

6338 

64n 

6483 

6556 

6629 

598 

67oi 

6774 

6846 

69i9 

6992 

7o64 

7i37 

7209 

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7354 

599 

7427 

7499 

7572 

7644 

7717 

7789 

-7862 

7934 

8006 

8o79 

72 

600 

8i5i 

8224 

8296 

8368 

844i 

85i3 

8585 

8658 

873o 

8802 

601 

8874 

8947 

9oi9 

9o9i 

9i63 

9236 

93o8 

938o 

9452 

9524 

602 

9596 

9669 

974i 

98i3 

9885 

9957 

..29 

.101 

,I73 

.245 

6o3 

78o3i7 

o389 

0461 

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o6o5 

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1181 

1253 

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i396 

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1612 

1  684 

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i755 

182-7 

i899 

1971 

2042 

2Il4 

2186 

2258 

2329 

24OI 

606 

2473 

2544 

2616 

2688 

2759 

283i 

29O2 

2974 

3o46 

3n7 

6o7 

3i89 

3260 

3332 

34o3 

3475 

3546 

36i8 

3689 

376i 

3832 

7i 

608 

39o4 

3975 

4o46 

4n8 

4i89 

4261 

4332 

44o3 

4475 

4546 

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46i7 

4689 

476o 

483i 

4902 

4974 

5o45 

5u6 

5i87 

5259 

610 

533o 

54oi 

5472 

5543 

56i5 

5686 

5757 

5828 

5899 

597o 

611 

6o4i 

6112 

6i83 

6254 

6325 

6396 

6467 

6538 

66o9 

6680 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

77   a5 

8 

i5 

23 

3i 

39 

46 

54 

62 

69 

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8 

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23 

3o 

38 

46 

53 

61 

68 

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75  PU 

8 

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23 

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38 

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53 

60 

68 

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74  ^< 

7 

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22 

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37 

44 

52 

59 

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22 

29 

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58 

66 

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22 

29 

36 

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58 

65 

71  * 

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21 

28 

36 

43 

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57 

64 

14 


LOGARITHMS    OF    J\UMBERS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

612 

786751 

6822 

6893 

6964 

7o35 

7106 

7i77 

7248 

73i9 

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71 

6i3 

746o 

753i 

7602 

7673 

7744 

78i5 

7885 

7956 

8o27 

8o98 

/ 

6i4 

8168 

8239 

83io 

838i 

845  1 

8522 

8593 

8663 

8734 

88o4 

616 

8875 

8946 

9016 

9087 

9l57 

9228 

9299 

9369 

944o 

95io 

616 

958i 

965i 

9722 

9792 

9863 

9933 

...4 

•  •74 

•  i44 

.215 

70 

617 

790285 

o356 

0426 

0496 

o567 

o637 

o7o7 

o778 

0848 

o9i8 

/ 

618 

0988 

io59 

1129 

1199 

1269 

i34o 

i4io 

i48o 

i55o 

1620 

619 

1691 

1761 

i83i 

1901 

1971 

2041 

2III 

2181 

2252 

2322 

620 

2392 

2462 

2532 

2602 

2672 

2742 

98l2 

2882 

2952 

3022 

€21 

3092 

3i62 

323i 

33oi 

3371 

344  1 

35u 

358i 

365i 

3721 

622 

379o 

386o 

3930 

4ooo 

4070 

4i39 

4209 

4279 

4349 

44i8 

628 

4488 

4558 

4627 

4697 

4767 

4836 

49o6 

4976 

5o45 

5n5 

624 

5i85 

5254 

5324 

5393 

5463 

5532 

56o2 

5672 

574i 

58n 

626 

588o 

5949 

6019 

6088 

6i58 

6227 

6297 

6366 

6436 

65o5 

69 

626 

6574 

6644 

6713 

6782 

6852 

6921 

699o 

7060 

7i29 

7198 

627 

7268 

7337 

7406 

7475 

?545 

76l4 

7683 

7752 

7821 

789o 

628 

7960 

8029 

8098 

8167 

8236 

83o5 

8374 

8443 

85i3 

8582 

629 

865i 

8720 

8789 

8858 

8927- 

8996 

9o65 

9i34 

9203 

0.272 

63o 
63i 

934i 
800029 

9409 
0098 

9478 
0167 

9547 
0236 

9616 
o3o5 

9685 
o373 

9754 
0442 

9823 

o5u 

9892 
o58o 

9961 
o648 

632 

0717 

0786 

o854 

0923 

0992 

1061 

II29 

1198 

1266 

i335 

633 

i4o4 

1472 

i54i 

i6o9 

1678 

1747 

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1  884 

I952 

2O2I 

634 

2089 

2i58 

2226 

2295 

2363 

2432 

25oo 

2568 

2637 

2705 

635 

2774 

2842 

2910 

2979 

3o47 

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3i84 

3252 

332i 

3389 

68 

636 

3457 

3525 

3594 

3662 

373o 

3798 

3867 

3935 

4oo3 

4071 

637 

4i39 

4208 

4276 

4344 

44i2 

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4548 

46i6 

4685 

4753 

638 

4821 

4889 

4957 

5o25 

5o93 

5i6i 

5229 

5297 

5365 

5433 

639 

55oi 

5569 

5637 

57o5 

5773 

584i 

59o8 

5976 

6o44 

6112 

64o 

6180 

6248 

63i6 

6384 

645  1 

65i9 

6587 

6655 

6723 

679o 

64i 

6858 

6926 

6994 

7061 

7I29 

7197 

7264 

7332 

74oo 

7467 

642 

7535 

7608 

7670 

7738 

7806 

7873 

794  1 

8008 

8o76 

8i43 

1  643 

8211 

8279 

8346 

84i4 

848  1 

8549 

8616 

8684 

875i 

8818 

67 

644 

8886 

8953 

9021 

9088 

9i56 

9223 

929o 

9358 

9425 

9492 

645 

956o 

9627 

9694 

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9829 

9896 

9964 

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.165 

646 

8io233 

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o77o 

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647 

0904 

0971 

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1106 

1173 

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1307 

i374 

i44i 

i5o8 

648 

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1642 

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1977 

2044 

2III 

2I78 

649 

2245 

2312 

2379 

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2579 

2646 

2713 

2780 

2847 

65o 

2913 

2980 

3o47 

3u4 

3i8i 

3247 

33i4 

338i 

3448 

35i4 

65: 

358i 

3648 

37i4 

378i 

3848 

3914 

398i 

4o48 

4n4 

4i8i 

652 

4248 

43i4 

438i 

4447 

45i4 

458i 

4647 

47i4 

4780 

4847 

653 

49i3 

4980 

5o46 

5n3 

5179 

5246 

53i2 

5378 

5445 

55u 

66 

654 

5578 

5644 

57n 

5777 

5843 

59io 

5976 

6042 

6io9 

6i75 

655 

6241 

63o8 

6374 

644o 

65o6 

6573 

6639 

67o5 

6771 

6838 

656 

6904 

6970 

7o36 

7102 

7169 

7235 

73oi 

7367 

7433 

7499 

657 

7565 

763i 

7698 

7764 

783o 

7896 

7962 

8028 

8o94 

8160 

658 

8226 

8292 

8358 

8424 

8490 

8556 

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8688 

8754 

8820 

659 

8885 

895i 

9017 

9o83 

9149 

92l5 

9281 

9346 

94l2 

9478 

660 

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9610 

5676 

9?4i 

9807 

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...4 

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661 

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663 

i5i4 

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i645 

1710 

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i84i 

1906 

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2037 

2103 

65 

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2168 

2233 

2299 

2364 

2430 

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2626 

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2756 

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0 

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59 

LOGARITHMS    OF    NUMBERS. 


N.    0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

665 

822822 

2887 

2952 

3oi8 

3o83 

3i48 

32i3 

3279 

3344 

34o9 

65 

666 

3474 

3539 

36o5 

3670 

3735 

38oo 

3865 

393o 

3996 

4o6i 

667 

4126 

4191 

4256 

4321 

4386 

445! 

45i6 

458i 

4646 

4711 

668 

4776 

484i 

49o6 

497i 

5o36 

5ioi 

5i66 

523i 

5296 

536i 

669 

5426 

5491 

5556 

562i 

5686 

575i 

58i5 

588o 

5945 

6010 

670 

6o75 

6i4o 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

671 

6723 

6787 

6852 

6917 

698i 

7046 

7x11 

7i75 

7240 

73o5 

672 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

•7821 

7886 

795i 

673 

8oi5 

8o'8o 

8i44 

8209 

8273 

8338 

8402 

8467 

853i 

8595 

64 

674 

8660 

8724 

8789 

8853 

89i8 

8982 

9046 

9in 

9i75 

9239 

675 

93o4 

9368 

9432 

9497 

956i 

9625 

9690 

9754 

9818 

9882 

676 

9947 

.  .  ii 

..75 

.i39 

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.396 

.46o 

.525 

677 

83o589 

o653 

o7i7 

0781 

0845 

o9o9 

0973 

io37 

IIO2 

1166 

678 

1230 

1294 

i358 

1422 

i486 

i55o 

i6i4 

16-78 

1742 

1806 

679 

i87o 

1934 

I998 

2062 

2126 

2l89 

2253 

23l7 

238i 

2445 

680 

25o9 

2573 

2637 

2700 

2764 

2828 

2892 

2956 

3O2O 

3o83 

681 

3i47 

3211 

3275 

3338 

3402 

3466 

353o 

3593 

3657 

3721 

• 

682 

3784 

3848 

39I2 

3975 

4o39 

4io3 

4i66 

4^3o 

4294 

4357 

683 

4421 

4484 

4548 

46ii 

4675 

4739 

4802 

4866 

4929 

4993 

684 

5o56 

5l2O 

5i83 

5247 

53io 

5373 

5437 

55oo 

5564 

5627 

63 

685 

569i 

5754 

58i7 

588i 

5944 

6007 

6o7i 

6r34 

6i97 

6261 

686 

6324 

6387 

645  1 

65i4 

6577 

664i 

6704 

6767 

683o 

6894 

687 

6957 

7020 

7o83 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

688 

7588 

7652 

77*5 

777s 

7841 

79°4 

7967 

8o3o 

8093 

8i56 

689 

8219 

8282 

8345 

84o8 

8471 

8534 

8597 

8660 

8723 

8786 

690 

8849 

8912 

8975 

9o38 

9ioi 

9164 

9227 

9289 

9352 

94i5 

691 

9478 

9541 

96o4 

9667 

9729 

9792 

9855 

99i8 

998i 

..43 

692 

84oio6 

0169 

0232 

0294 

o357 

O42O 

0482 

o545 

0608 

0671 

693 

o733 

0796 

o859 

0921 

0984 

io46 

no9 

1172 

1234 

I297 

694 

.  i359 

1422 

i485 

i547 

1610 

1672 

i735 

1797. 

1860 

I922 

695 

i985 

2047 

2IIO 

2172 

2235 

2297 

236o 

2422 

2484 

2547 

6£ 

696 

26o9 

2672 

2734 

2796 

2859 

2921 

2983 

3o46 

3io8 

3170 

697 

3233 

3295 

3357 

3420 

3482 

3544 

36o6 

3669 

373i 

3793 

698 

3855 

39i8 

398o 

4042 

4io4 

4i66 

4229 

4291 

4353 

44i5 

699 

4477 

4539 

46oi 

4664 

4726 

4788 

485o 

4912 

4974 

5o36 

700 

5o98 

5i6o 

5222 

5284 

5346 

54o8 

547o 

5532 

5594 

5656 

701 

'  57i8 

5780 

5842 

59o4 

5966 

6028 

6o9o 

6i5i 

6213 

6275 

702 

6337 

6399 

646  1 

6523 

6585 

6646 

67o8 

6770 

6832 

6894 

703 

6955 

7017 

7079 

7141 

7202 

•7264 

7326 

7388 

7449 

75n 

704 

7573 

7634 

7696 

7758 

78i9 

788i 

7943 

8oo4 

8066 

8128 

7o5 

8i89 

825i 

83i2 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

706 

88o5 

8866 

8928 

8989 

9o5i 

9112 

9i74 

9235 

9297 

9358 

61 

707 

94i9 

948i 

9542 

9604 

9665 

9-726 

9788 

9849 

9911 

9972 

708 

85oo33 

oo95 

oi56 

0217 

0279 

o34o 

o4oi 

0462 

o585 

7°9 

o646 

0707 

0769 

o83o 

o89i 

0952 

1014 

1075 

n36 

1197 

710 

1258 

1320 

i38i 

1  442 

i5o3 

1  564 

i625 

1686 

1747 

1809 

711 

i87o 

I93i 

I992 

2o53 

2114 

2I75 

2236 

2297 

2358 

2419 

712 

2480 

254i 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

7i3 

3o9o 

3i5o 

3211 

3272 

3333 

3394 

3455 

35i6 

3577 

3637 

7i4 

3698 

3759 

3820 

388i 

394i 

4OO2 

4o63 

4i24 

4i85 

4245 

7i5 

43o6 

4367 

4428 

4488 

4549 

46io 

467o 

473i 

4792 

4852 

716 

49i3 

4974 

5o34 

5o95 

5i56 

52i6 

5277 

5337 

5398 

5459 

71-7 

55i9 

558o 

564o 

5701 

576i 

5822 

5882 

5943 

6oo3 

6o64 

718 

6124 

6i85 

6245 

63o6 

6366 

642-7 

6487 

6548 

6608 

6668 

60 

7i9 

6729 

6789 

685o 

69io 

697o 

7o3i 

7o9i 

7i52 

7212 

7272 

i  N' 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

f  64  » 

6 

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26 

32 

38 

45 

5i 

58 

±   63  S 

6 

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25 

32 

38 

44 

5o 

57 

£\  62  PH  . 

6 

12 

25 

3i 

37 

43 

5o 

56 

5   61  g 

6 

12 

18 

24 

3i 

43 

49 

55 

I  60  p., 

6 

12 

18 

24 

3o 

36 

42 

48 

54 

! 

LOGARITHMS    OF    NUMBERS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9  1  D. 

720 

857332 

7393 

7453 

75i3 

7b74 

7634  '  7694 

7755 

7815 

7875 

60  f 

721 

7935 

7995 

8o56 

8116 

8i76 

8236 

8297 

8357 

8417 

8477 

722 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9o78 

728 

9i38 

9198 

9258 

93i8 

9379 

9439 

9499 

9559 

9619 

9679 

724 

9739 

9799 

9859 

9918 

9978 

..38 

..98 

.i58 

.218 

.278 

726 

86o338 

0398 

o458 

o5i8 

o578 

o637 

0697 

o757 

0817 

o877 

726 

o937 

0996 

io56 

1116 

n76 

1236 

1295 

i355 

i4i5 

i475 

727 

i534 

i594 

i654 

1714 

1773 

i833 

1893 

1952 

2OI2 

20-72 

728 

2l3l 

2191 

225l 

23lO 

2370 

243o 

2489 

2549 

2608 

2668 

729 

2728 

2787 

2847 

2906 

2966 

3o25 

3o85 

3i44 

3204 

3263 

780 

3323 

3382 

3442 

35oi 

356i 

3620 

368o 

3739 

3-799 

3858 

59 

73i 

39i7 

3977 

4o36 

4096 

4i55 

42i4 

4274 

4333 

4392 

4452 

732 

45n 

457o 

463o 

4689 

4743 

48o8 

4867 

4926 

4985 

5o45 

733 

5io4 

5i63 

5222 

5282 

534i 

54oo 

5459 

55i9 

5578 

5637 

734 

5696 

5755 

58i4 

5874 

5933 

5992 

6o5! 

6110 

6169 

6228 

735 

6287 

6346 

64o5 

6465 

6524 

6583 

6642 

67oi 

676o 

6819 

736 

6878 

693-7 

6996 

7o55 

7ii4 

7i73 

7232 

729I 

735o 

74o9 

73? 

7467 

7526 

7585 

7644 

77°3 

7762 

7821 

7880 

7939 

7998 

738 

8o56 

8n5 

8i74 

8233 

8292 

835o 

8409 

8468 

8527 

8586 

739 

8644 

87o3 

8762 

8821 

8879 

8938 

8997 

9o56 

9114 

9i73 

74o 

9282 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

97oi 

976o 

74  1 

98i8 

9877 

9935 

9994 

..53 

.in 

.  I7O 

.228 

.28-7 

.345 

742 

87o4o4 

0462 

0521 

o579 

o638 

0696 

o755 

o8i3 

0872 

ogSo 

58 

743 

o989 

1047 

1106 

n64 

1223 

1281 

1339 

1398 

i456 

i5i5 

744 

:573 

i63i 

1690 

i748 

l8o6 

i865 

1923 

1981 

2O40 

2098 

745 

2166 

22l5 

2273 

233i 

2389 

2448 

25o6 

2564 

2622 

2681 

746 

2739 

2797 

2855 

2913 

2972 

3o3o 

3o88 

3i46 

3204 

3262 

747 

332i 

3379 

3437 

3495 

3553 

36n 

3669 

3727 

3785 

3844 

748 

3902 

3960 

4oi8 

4o76 

4i34 

4192 

425o 

43o8 

4366 

4424 

749 

4482 

4540 

4598 

4656 

47i4 

4772 

483o 

4888 

4945 

5oo3 

75o 

5o6i 

Siig 

5i77 

5235 

5293 

535i 

5409 

5466 

5524 

5582 

75l 

564o 

5698 

5756 

58i3 

587i 

5929 

5987 

6o45 

6102 

6160 

752 

6218 

6276 

6333 

6391 

6449 

65o7 

6564 

6622 

6680 

6737 

753 

6795 

6853 

6910 

6968 

•7026 

7o83 

7i4i 

7199 

7256 

73i4 

754 

737i 

7429 

7487 

7544 

•7602 

7659 

77i7 

7774 

7832 

-7889 

755 

7947 

8oo4 

8062 

8119 

8177 

8234 

8292 

834g 

84o7 

8464 

57 

756 

8622 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

757 

9o96 

9i53 

9211 

9268 

9325 

9383 

944o 

9497 

9555 

9612 

768 

9669 

9726 

9784 

984i 

q898 

9956 

..i3 

..70 

.127 

.i85 

759 

880242 

0299 

o356 

o4i3 

o47i 

0028 

o585 

O642 

0699 

o756 

760 

o8i4' 

o87i 

0928 

0985 

1042 

1099 

n56 

I2l3 

I27I 

i328 

76i 

i385 

1  442 

1499 

i556 

i6i3 

i67o 

I727 

1784 

i84i 

1898 

762 

i955 

2OI2 

2069 

2126 

2i83 

2240 

2297 

2354 

241  1 

2468 

763 

2626 

258l 

2638 

2696 

2752 

2809 

2866 

2923 

2980 

3o37 

764 

3o93 

3i5o 

3207 

3264 

332i 

3377 

3434 

3491 

3548 

36o5 

765 

366i 

37i8 

3775 

3832 

3888 

3945 

40O2  ' 

4o5g 

4n5 

4l72 

766 

4229 

4285 

434s 

4399 

4455 

45i2 

4569 

4625 

4682 

4739 

767 

4795 

4852 

4909 

4g65 

5O22 

5o78 

5i35 

5192 

5248 

53o5 

768 

536i 

54i8 

5474 

553i 

5587 

5644 

57oo 

5757 

58i3 

587o 

769 

5926 

5983 

6039 

6096 

6i52 

6209 

6265 

632i 

6378 

6434 

56 

77° 

649i 

6547 

66o4 

6660 

67i6 

6773 

6829 

6885 

6942 

6998 

771 

7o54 

7111 

7i67 

7223 

7280 

7336 

7392 

7449 

75o5 

756i 

7-72 

7617 

7674 

773o 

7786 

7842 

7898. 

7955 

8011 

8o67 

8i23 

773 

8l79 

8236 

8292 

8348 

84o4 

846o 

85i6 

8573 

8629 

8685 

774 

874i 

8797 

8853 

8909 

8965 

9021 

9°77 

9i34 

9190 

9246 

N. 

0   |  1. 

2 

3 

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5 

6 

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60  «  r  6 

12 

18 

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3o 

36 

42 

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LOGARITHMS    of    NUMBERS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9  1  D_] 

775 

889802 

9358 

94i4 

9470 

9626 

9582 

9638 

9b94 

975o 

98o6 

5fi 

776 

9862 

9918 

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..86 

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•197 

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.309 

.365 

777 

890421 

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o533 

0589 

o645 

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0812 

0868 

o924 

778 

0980 

io35 

1091 

1  1  47 

1203 

1259 

i3i4 

1370 

1426 

1482 

779 

1637 

1593 

1649 

1705 

i76o 

1816 

l872 

I928 

1983 

2o39 

780 

2095 

2i5o 

2206 

2262 

23l7 

2373 

2429 

2484 

254o 

a595 

781 

2661 

2707 

'2*762 

2818 

2873 

2929 

2985 

3o4o 

3096 

3i5i 

782 

3207 

3262 

33i8 

3373 

3429 

3484 

354o 

3595 

365i 

37o6 

783 

3762 

38i7 

3873 

3928 

3984 

4o39 

4og4 

4i5o 

42o5 

4261 

55 

784 

43i6 

437i 

442  7 

4482 

4538 

4593 

4648 

47°4 

4759 

48i4 

7S5 

4870 

4925 

4980 

5o36 

5091 

5i46 

52OI 

5257 

53i2 

5367 

786 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

58o9 

5864 

5920 

787 

5975 

6o3o 

6o85 

6i4o 

6195 

625i 

63o6 

636i 

64i6 

647i 

788 

6626 

658i 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

789 

7077 

7l32 

7187 

724.2 

7297 

735a 

74o7 

7462 

75i7 

7572 

79° 

7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8o67 

8122 

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791 

8176 

823i 

8286 

834i 

8396 

845  1 

85o6 

856i 

86i5 

867o 

792 

8726 

878o 

8835 

8890 

8944 

8999 

9o54 

9109 

9i64 

92l8 

793 

0273 

9828 

9383 

9437 

9492 

9547 

96o2 

9656 

9711 

9766 

794 

9821 

9875 

993o 

9985 

..39 

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.i49 

.2O3 

.258 

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79  5 

9oo367 

0422 

o476 

o53i 

o586 

o64o 

o695 

0749 

0804 

o859 

7-96 

0913 

0968 

1022 

io77 

n3i 

1186 

1240 

1295 

i349 

i4o4 

797 

i458 

i5i3 

i567 

1622 

i676 

I73i 

i785 

i84o 

i894 

i948 

54 

798 

20O3 

2o57 

2112 

2166 

2221 

2276 

2329 

2384 

2438 

2492 

799 

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2601 

2655 

27IO 

2764 

2818 

2873 

2927 

298l 

3o36 

800 

3090 

3i44 

3199 

3253 

33o7 

336i 

34i6 

3470 

3524 

3578 

801 

3633 

3687 

374i 

3795 

3849 

39o4 

3958 

4012 

4o66 

4l2O 

802 

4i74 

4229 

4283 

4337 

439i 

4445 

4499 

4553 

46o7 

466i 

8o3 

4716 

477o 

4824 

4878 

4g32 

4986 

5o4o 

5o94 

5i48 

52O2 

8o4 

6266 

53io 

5364 

54i8 

5472 

5526 

558o 

5634 

5688 

5742 

8o5 

6796 

585o 

5904 

5958 

6012 

6066 

6n9 

6i73 

6227 

6281 

806 

6335 

6389 

6443 

6497 

655i 

66o4 

6658 

6712 

6766 

6820 

807 

6874 

692  7 

6981 

7o35 

7089 

7i43 

7i96 

725o 

73o4 

7358 

j 

808 

74.11 

7465 

75i9 

7573 

7626 

7680 

7734 

7787 

784i 

7895 

809 

7949 

8002 

8o56 

8110 

8i63 

8217 

82-70 

8324 

8378 

843i 

8xo 

8485 

8539 

8592 

8646 

8699 

8753 

88o7 

8860 

89i4 

8967 

8n 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

95o3 

812 

9556 

9610 

9663 

9716 

977° 

9823 

9877 

993o 

9984 

..37 

53 

8i3 

910091 

oi44 

0197 

025l 

o3o4 

o358 

o4n 

o464 

o5i8 

o57I 

8i4 

0624 

0678 

0781 

0784 

o838 

o89i 

o944 

o998 

io5i 

no4 

8x5 

n58 

I2II 

1264 

1317 

1871 

i424 

1477 

i53o 

i584 

i637 

816 

1690 

1743 

1797 

i85o 

I9o3 

i956 

2009 

2o63 

2116 

2i69 

817 

2222 

2275 

2328 

238i 

2435 

2488 

254i 

2594 

2647 

27OO 

818 

2753 

2806 

2859 

2913 

2966 

3oi9 

3o72 

3i25 

3i78 

323i 

819 

3284 

3337 

3390 

3443 

3496 

3549 

36o2 

3655 

37o8 

376i 

820 

38:4 

3867 

3920 

3973 

4026 

4o79 

4l32 

4:84 

4237 

4290 

821 

4343 

4396 

4449 

45o2 

4555 

46o8 

466o 

47i3 

4766 

48i9 

822 

4872 

4925 

4977 

5o3o 

5o83 

5i36 

5i89 

524i 

5294 

5347 

823 

54oo 

5453 

55o5 

5558 

56n 

5664 

57i6 

5769 

5822 

5875 

824 

5927 

5g8o 

6o33 

6o85 

6i38 

6i9i 

6243 

6296 

6349 

64oi 

825 

6454 

65o7 

6559 

6612 

6664 

67i7 

677o 

6822 

6875 

6927 

826 

6980 

7o33 

7o85 

7i38 

7190 

7243 

7295 

7348 

74oo 

7453 

827 

75o6 

7558 

76n 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

52 

828 

8o3o 

8o83 

8i35 

8188 

8240 

8293 

8345 

8397 

845o 

85o2 

829 

8555 

86o7 

8659 

87I2 

8764 

8816 

8869 

892I 

8973 

9026 

83o 

9078 

9i3o 

9183 

9235 

9287 

934o 

9392 

9444 

9496 

9549 

N. 

0 

1  |  2 

3 

4   |1  5 

6 

7 

8 

9 

D. 

r  55  « 

6 

ii 

1.7 

22 

28 

33 

39 

44 

5o 

|J  54  S 

5 

ii 

16 

22 

27 

32 

38 

43 

49 

|]  53  * 

5 

ii 

16 

21 

27 

32 

37 

42- 

48 

"•*  I  52  p^ 

5 

10 

16 

21 

26 

3i 

36 

42 

47 

| 

18 


LOGARITHMS    OF    NUMBERS. 


^r 

•  o  |  i 

2    3 

4     5 

6  |  7 

8 

9 

D-J 

,  83i 

919601  •  g653  9706 

9758 

9810 

9862 

9914 

9967 

..19 

..71 

HTJ 

832 

920123 

0176 

0228 

0280 

o332 

o384 

o436 

0489 

o54i 

o593 

I 

833 

o645 

0697 

o749 

0801 

o853 

0906 

0958 

IOIO 

1062 

in4 

834 

1166 

isTs 

1270 

1322 

i374 

1426 

1478 

T53o 

i58s 

1  634 

835 

1686 

i738 

1790 

1842 

i894 

1946 

1998 

2o5o 

2102 

2i54 

836 

2206 

2258 

a3io 

2362 

2414 

2466 

25i8 

257O 

2622 

2674 

837  i   2725 

2777 

2829 

2881 

2933 

2985 

3o37 

3o89 

3i4o 

3i92 

838 

3244 

3296 

3348 

3399 

345  1 

35o3 

3555 

36o7 

3658 

37io 

839 

376a 

38i4 

3865 

39i7 

3969 

4021 

4072 

4i24 

4176 

4228 

84o 

4279 

433i 

4383 

4434 

4486 

4538 

4589 

464i 

4693 

4744 

84  1 

4796 

4848 

4899 

495i 

5oo3 

5o54 

5io6 

5i57 

5209 

5261 

842 

53ia 

5364 

54i5 

5467 

55i8 

5570 

562i 

5673 

5?25 

5776 

843 

5828 

5879 

593i 

5982 

6o34 

6o85 

6i37 

6188 

6240 

629i 

5i 

844 

634a 

6394 

6445 

6497 

6548 

6600 

665i 

67O2 

6754 

68o5 

845 

6857 

69o8 

6959 

7011 

7062 

7u4 

7i65 

72l6 

7268 

73i9 

846 

737o 

7422 

7473 

7524 

7576 

7627 

7678 

773o 

778i 

7832 

84? 

7883 

7935 

7986 

8o37 

8088 

8i4o 

8191 

8242 

8293 

8345 

848 

8396 

8447 

8498 

8549 

8601 

8652 

87o3 

8754 

88o5 

8857 

849 

8908 

8959 

9010 

9061 

9II2 

9163 

92l5 

9266 

93i7 

9368 

85o 

9419 

9470 

9521 

9072 

9623 

9674 

9725 

9776 

9827 

9879 

85i 

9930 

9981 

..32 

..83 

.134 

.186 

.236 

.287 

.338 

.889 

862 

93o44o 

0491 

o542 

0592 

o643 

0694 

o745 

0796 

0847 

0898 

853 

0949 

1000 

io5i 

IIO2 

n53 

1204 

1254 

i3o5 

i356 

i4o7 

854 

i458 

1509 

i56o 

1610 

1661 

I7I2 

i763 

1814 

i865 

1915 

855 

1966 

2017 

2068 

2118 

2l69 

2220 

227I 

2322 

2372 

2423 

856 

2474' 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

857 

2981 

3o3i 

3o82 

3i33 

3i83 

3234 

3285 

3335 

3386 

3437 

858 

3487 

3538 

3589 

3639 

369o 

374o 

379i 

384i 

3892 

3943 

859 

3993 

4o44 

4094 

4i45 

4i95 

4246 

4296 

4347 

4397 

4448 

860 

4498 

4549 

4599 

465o 

4700 

4?5i 

48oi 

4852 

4902 

4953 

5o 

861 

5oo3 

5o54 

5io4 

5i54 

52o5 

5255 

53o6 

5356 

54o6 

5457 

862 

55o7 

5558 

56o8 

5658 

5709 

5759 

58og 

586o 

59io 

5960 

) 

863 

6011 

6061 

6m 

6162 

6212 

6262 

63i3 

6363 

64i3 

6463 

864 

65i4 

6564 

66i4 

6665 

6715 

6765 

68i5 

6865 

69i6 

6966 

865 

7016 

7066 

7117 

7167 

7217 

7267 

73i7 

7367 

74i8 

7468 

866 

75i8 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

•7969 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

837o 

8420 

847o 

868 

8520 

857o 

8620 

8670 

8720 

877o 

8820 

8870 

8920 

•8970 

869 

9020 

9070 

9120 

9170 

9220 

92-70 

9320 

9369 

9419 

9469 

870 

95i9 

9569 

9619 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

871 

940018 

0068 

0118 

0168 

0218 

O267 

o3i7 

o367 

0417 

o467 

872 

o5i6 

o566 

0616 

0666 

0716 

o765 

o8i5 

o865 

0915 

0964 

873 

ioi4 

io64 

ii  i4 

n63 

I2l3 

1263 

i3i3 

1  362 

1412 

1462 

874 

i5n 

i56i 

1611 

1660 

1710 

i76o 

1809 

i859 

1909 

i958 

875 

2008 

2o58 

2107 

2157 

2207 

2256 

23o6 

2355 

24o5 

2455 

876 

25o4 

2554 

26o3 

2653 

2702 

2752 

2801 

285i 

2901 

295o 

877 

3ooo 

3o49 

3o99 

3i48 

3198 

3247 

3297 

3346 

3396 

3445 

49 

878 

3495 

3544 

3593 

3643 

3692 

3742 

379i 

384i 

3890 

3939 

879 

3989 

4o38 

4o88 

4i37 

4i86 

4236 

4285 

4335 

4384 

4433 

880 

4483 

4532 

458i 

463i 

468o 

4729 

4779 

4828 

4877 

4927 

881 

4976 

5o25 

5o74 

5i24 

5i73 

5222 

5272 

532i 

537o 

54i9 

882 

5469 

55i8 

5567 

56i6 

5665 

57i5 

5764 

58i3 

5862 

5912 

883 

596i 

6010 

6o59 

6108 

6i57 

6207 

6256 

63o5 

6354 

64o3 

884 

6452 

65oi 

655i 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

885 

6943 

6992 

7041 

7090 

7i4o 

7189 

7238 

7287 

7336 

7385 

886 

7434 

7483 

7532 

758i 

763o 

7679 

7728 

7777 

7826 

7875 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

{"  62  a  f  5 

10 

16 

21 

26 

3i 

36 

42 

47 

5i  S  1   5 

10 

i5 

2O 

26 

3i 

36 

4i 

46 

5o  &0   5 

IO 

i5 

2O 

25 

3o 

35 

4o 

45 

49  •  *  i  5 

10 

i5 

2O 

25 

29 

34 

39 

44 

LOGARITHMS    OF    NUMBERS. 


N. 

0 

1 

o 

3 

4  I   5 

6 

7 

8 

9 

D. 

f  88? 

947924 

7973 

8022 

8070 

8119 

8168  82i7 

8266 

83i5 

8364 

49 

888 

84i3 

8462 

85n 

856o 

8609 

8657 

8706 

8755 

88o4 

8853 

1  889 

8902 

895i 

8999 

9o48 

9097 

9146 

9i95 

9244 

9292 

934i 

890 

9390 

9439 

9488 

9536 

9585 

9634 

9683 

973i 

978o 

9829 

891 

9878 

9926 

9975 

..24 

..73 

.121 

.  170 

.2I9 

.267 

.3i6 

892 

95o365 

04  1  4 

0462 

o5n 

o56o 

O6o8 

o657 

o7o6 

o754 

o8o3 

893 

o85i 

0900 

o949 

°997 

io46 

io95 

n43 

II92 

1240 

1289 

894 

i338 

i386 

i435 

i483 

i532 

i58o 

i629 

i677 

I726 

i775 

895 

1823 

1872 

I92O 

i969 

2017 

2066 

2Il4 

2i63 

2211 

2260 

48 

896 

23o8 

2356 

24o5 

2453 

2502 

255o 

2599 

2647 

2696 

2744 

597 

2792 

2841 

2889 

2938 

2986 

3o34 

3o83 

3i3i 

3i8o 

3228 

898 

3276 

3325 

3373 

3421 

34?0 

35i8 

3566 

36i5 

3663 

8711 

899 

376o 

38o8 

3856 

39o5 

3953 

4ooi 

4o4g 

4o98 

4i46 

4194 

900 

4243 

4291 

4339 

4387 

4435 

4484 

4532 

458o 

4628 

4677 

901 

4726 

4??3 

4821 

4869 

4918 

4966 

5oi4 

5o62 

5no 

5i58 

902 

6207 

5255 

53o3 

535i 

5399 

5447 

5495 

5543 

5592 

564o 

903 

5688 

5736 

5784 

5832 

588o 

5928 

5976 

6024 

6o72 

6120 

904 

6168 

6216 

6265 

63i3 

636i 

6409 

645  7 

65o5 

6553 

6601 

906 

6649 

6697 

6745 

6793 

684o 

6888 

6936 

6984 

7o32 

7o8o 

906 

7128 

7176 

7224 

7272 

7320 

7368 

74i6 

7464 

75l2 

7559 

907 

7607 

7655 

77o3 

775i 

7799 

7847 

7894 

794a 

799° 

8o38 

908 

8086 

8i34 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

85i6 

909 

8564 

8612 

8659 

8707 

8755 

88o3 

885o 

8898 

8946 

8994 

910 

9041 

9089 

9i37 

9i85 

9232 

9280 

9328 

9375 

9423 

947i 

911 

95i8 

9566 

96i4 

966i 

97°9 

9757 

98o4 

9852 

9900 

9947 

912 

9995 

..42 

.  .9o 

.i38 

.i85 

.233 

.280 

.328 

.376 

.423 

918 

960471 

o5i8 

o566 

o6i3 

0661 

o7o9 

0756 

0804 

o85i 

0899 

914 

0946 

0994 

io4i 

io89 

ii36 

n84 

I23l 

I279 

i326 

i374 

47 

9i5 

1421 

1469 

i5i6 

i563 

1611 

i658 

1706 

I753 

1801 

1  848 

916 

1895 

1943 

I99o 

2038 

2085 

2l32 

2180 

2227 

2275 

2322 

917 

2369 

2417 

2464 

25ll 

2559 

2606 

2653 

2701 

2  748 

2795 

918 

2843 

2890 

2937 

2985 

3o32 

3o79 

3i26 

3i74 

3221 

3268 

919 

33i6 

3363 

34io 

3457 

35o4 

3552 

3599 

3646 

3693 

374i 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4n8 

4i65 

4212 

921 

4260 

43o7 

4354 

44oi 

4448 

4495 

4542 

459o 

4637 

4684 

922 

4?3i 

4778 

4825 

4872 

4919 

4966 

5oi3 

5o6i 

5io8 

5i55 

923 

52O2 

5249 

5296 

5343 

539o 

5437 

5484 

553i 

5578 

5625 

924 

6672 

5719 

5766 

58:3 

586o 

59o7 

5954 

6001 

6o48 

6095 

925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

65i7 

6564 

926 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7o33 

927 

7080 

7127 

7i73 

7220 

7267 

73i4 

736i 

74o8 

7454 

75oi 

928 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

•7969 

929 

8016 

8062 

8io9 

8i56 

8203 

8249 

8296 

8343 

839o 

8436 

93o 

8483 

853o 

8576 

8623 

8670 

87i6 

8763 

8810 

8856 

89o3 

93i 

895o 

8996 

9o43 

9090 

9i36 

9i83 

9229 

9276 

9323 

9369 

932 

9416 

9463 

95oo 

9556 

96o2 

9649 

9695 

9742 

9789 

9835 

933 

9882 

9928 

9975 

.  .21 

..68 

.114 

.161 

.2O7 

.254 

.3oo 

934 

970347 

o393 

o44o 

o486 

o533 

o579 

0626 

o672 

°7T9 

o765 

46 

935 

0812 

o858 

o9o4 

095  1 

o997 

io44 

io9o 

n37 

n83 

I229 

936 

1276 

1322 

i369 

i4i5 

i46i 

i5o8 

1  554 

1601 

i647 

i693 

937 

1740 

I786 

i832 

1879 

I925 

i97i 

2018 

2064 

21  IO 

2l57 

938 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

26l9 

939 

2666 

2712 

2758 

2804 

285i 

2897 

2943 

2989 

3o35 

3082 

940 

3128 

3i74 

322O 

3266 

33i3 

3359 

34o5 

345  1 

3497 

3543 

941 

SSgo 

3636 

3682 

3728 

3774 

3820 

3866 

39i3 

3959 

4oo5 

942 

4o5i 

4097 

4i43 

4189 

4235 

4281 

4327 

4374 

44so 

4466 

943 

45i2 

4558 

46o4 

465o 

4696 

4742 

4788 

4834 

488o 

4926 

N.  |   0     1 

2  , 

3 

4   ||  5 

6 

7 

8 

9 

D. 

g  f  48  «  f  5 

10 

i4 

19    24 

29 

34 

38 

43 

1  {  47  *  1   5 

9   i4 

19    24 

28 

33 

38 

42 

0  1  46  £  (  5 

9   :4 

18     23 

28 

32 

37 

4i 

— 1 


LOGARITHMS    OF    NUMBERS 


N. 

0 

1 

2 

3 

4 

5 

6 

; 

8 

9 

D 

944 

974972 

6018 

5o64 

5no 

5i56 

5202 

5248 

5294 

534o 

5386 

46 

945 

5432 

5478 

5524 

557o 

56i6 

5662 

5707 

5753 

5799 

5845 

946 

589i 

5937 

5983 

6o29 

6o75 

6121 

6167 

6212 

6a58 

63o4 

947 

635o 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

67i7 

6763 

948 

6808 

6854 

69oo 

6946 

6992 

7o37 

7083 

7I29 

7i75 

7220 

949 

7266 

7312 

7358 

74o3 

7449 

7495 

754i 

7586 

7632 

7678 

o5o 

7724 

7769 

78i5 

7861 

•7906 

795a 

7998 

8o43 

8089 

8i35 

95i 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

85oo 

8546 

859i 

062 

8687 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9o47 

953 

9o93 

9i38 

9i84 

923o 

9275. 

9821 

9366 

94ia 

9457 

95o3 

954 

9548 

9594 

9639 

9685 

973o 

9776 

982I 

9867 

9912 

9958 

966 

980008 

oo,  ty 

oo94 

oi4o 

oi85 

023l 

0276 

0322 

o367 

O4l2 

45 

966 

o458 

o5o3 

o549 

o594 

o64o 

o685 

0780 

o776 

0821 

0867 

967 

09I2 

0967 

ioo3 

io48 

io93 

1139 

1184 

1229 

I275 

1320 

958 

i366 

i4n 

i456 

i5oi 

1  547 

i592 

i637 

i683 

1-728 

i773 

969 

1819 

i864 

I9°9 

i954 

2OOO 

ao45 

2O9O 

2i35 

2181 

2226 

960 

2271 

a3i6 

2362 

2407 

2452 

2497 

2543 

a588 

a633 

2678 

961 

2728 

2769 

2814 

2859 

2904 

2949 

2994 

3o4o 

3o85 

3i3o 

962 

3i75 

3220 

3265 

33io 

3356 

34oi 

3446 

3491 

3536 

358i 

968 

3626 

367i 

37i6 

3762 

3807 

3852 

3897 

3942 

3987 

4o32 

964 

4077 

4l22 

4i67 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

965 

4627 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

966 

4977 

5022 

5067 

5lI2 

5i57 

52O2 

5247 

5292 

5337 

5382 

967 

5426 

5471 

55i6 

556i 

56o6 

565i 

5696 

574i 

5786 

583o 

968 

5875 

3^20 

5965 

6010 

6o55 

6100 

6i44 

6189 

6234 

6279 

969 

6324 

6369 

64i3 

6458 

65o3 

6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

69o6 

695i 

6996 

7o4o 

7o85 

7i3o 

7i75 

971 

7219 

7264 

73o9 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

973 

8n3 

8157 

8202 

8247 

829I 

8336 

838i 

8425 

847o 

85i4 

974 

8559 

86o4 

8648 

8693 

8737 

8782 

8826 

887i 

8916 

896o 

975 

9006 

9o49 

9o94 

9i38 

9i83 

9227 

9272 

93i6 

936i 

94o5 

976 

9460 

9494 

9539 

9583 

9628 

9672 

9717 

976i 

98o6 

985o 

44 

977 

9895 

9939 

9983 

..28 

..72 

.117 

.161 

.206 

.25o 

.294 

978 

99o339 

o383 

0428 

0472 

o5i6 

o56i 

o6o5 

o65o 

o694 

o738 

979 

0783 

0827 

0871 

o9i6 

o96o 

ioo4 

1049 

1093 

n37 

1182 

980 

1226 

1270 

i3i5 

i359 

i4o3 

i448 

l492 

i536 

i58o 

1625 

981 

1669 

1718 

1758 

1802 

i846 

1890 

i935 

19-79 

2O23 

2o67 

982 

21  I  I 

2i56 

22OO 

2244 

2288 

2333 

2377 

2421 

2465 

25o9 

983 

2554 

2598 

2642 

2686 

273o 

2774 

28l9 

2863 

2907 

295i 

984 

2995 

3o39 

3o83 

8127 

3l72 

32i6 

3260 

33o4 

3348 

3392 

985 

3436 

348o 

3524 

3568 

36i3 

3657 

3701 

3745 

3789 

3833 

986 

3877 

392I 

3965 

4oo9 

4o53 

4097 

4i4i 

4i85 

4229 

4273 

987 

43i7 

436i 

44o5 

4449 

4493 

4537 

458i 

4625 

4669 

47i3 

988 

4757 

48oi 

4845 

4889 

4933 

4977 

5O2I 

5o65 

5io8 

5i52 

989 

5i96 

524o 

5284 

5328 

5372 

54i6 

5460 

55o4 

5547 

559i 

99° 

5635 

5679 

5723 

5767 

58n 

5854 

5898 

5942 

5986 

6o3o 

991 

6074 

6117 

6161 

62o5 

6249 

6293 

6337 

638o 

64^4 

6468 

992 

65i2 

6555 

6599 

6643 

6687 

6781 

6774 

6818 

6862 

69o6 

993 

6949 

6993 

7o37 

7o8o 

7124 

7168 

7212 

7255 

7299 

7343 

994 

7386 

743o 

7474 

75i7 

756i 

7605 

7648 

7692 

7736 

7779 

996 

7823 

7867 

79io 

7954 

7998 

8o4i 

8o85 

8l29 

8172 

8216 

996 

8259 

83o3 

8347 

839o 

8434 

8477 

852i 

8564 

8608 

8652 

997 

8695 

8739 

8782 

8826 

8869 

89i3 

8956 

9ooo 

9043 

9o87 

998 

9i3i 

9i74 

92l8 

926l 

93o5 

9348 

9392 

9435 

9479 

9522 

999 

9565 

96o9 

9652 

9696 

9739 

9783 

9826 

987o 

99i3 

9957 

43 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

.  f  46  a  f   5 

9 

i4 

18 

23 

28 

32 

37 

4i 

J   45  S  I   5 

9 

i4 

18 

23 

27 

32 

36 

4i 

§   44  M   4 

9 

i3 

18 

22 

26 

3i 

35 

4o 

Q  I  43  p;  [  4 

9 

i3 

J7 

22 

26 

3o 

34 

39 

J 

TABLE 


OP 


LOGARITHMIC  SUES  AID  TANGENTS 


FOR    EVERY 


TEN  SECONDS  OF  THE  QUADRANT. 


LOGARITHMIC    SINES. 


Win. 

Sine  of  0  Degree. 

P.  Part 

0" 

10" 

20" 

30" 

40" 

50" 

to  1". 

O 

Inf.  Neg. 

5.685575 

5.9866o5 

6.162696 

6.287635 

6.384545 

59 

I 

6.463726 

6.53o673 

6.588665 

639817 

685575 

726968 

58 

2 

764756 

7995i8 

83i7o3 

861666 

889695 

916024 

57 

3 

940847 

964328 

986605 

7.007794 

7.027997 

7-o473o3 

56 

4 

7.065786 

7.o835i5 

7.  ioo548 

116939 

i32733 

147973 

55 

5 

162696 

176936 

190725 

204089 

217054 

229643 

54 

6 

241877 

253776 

265358 

276639 

287635 

298358 

53 

7 

308824 

319043 

329027 

338787 

348332 

357672 

52 

8 

3668i6 

37577i 

384544 

393i45 

401578 

409850 

5i 

9 

417968 

425937 

43376a 

44i449 

449002 

456426 

5o 

10 

463726 

470904 

477966 

4849  i  5 

491754 

498488 

49 

689.4 

ii 

5o5n8 

5n649 

5i8o83 

524423 

530672 

536832 

48 

629.4 

19 

542906 

548897 

5548o6 

56o635 

566387 

572o65 

47 

579.1 

i3 

577668 

583201 

588664 

5g4o59 

599388 

6o4652 

46 

536.2 

i4 

6o9853 

614993 

620072 

625093 

63oo56 

634964 

45 

499.2 

i5 

6398i6 

6446i5 

64936i 

654o56 

658701 

663297 

44 

467.o 

16 

667845 

672345 

676799 

681208 

685573 

689895 

43 

438.7 

i? 

69417? 

698410 

702606 

706762 

710879 

714957 

42 

4i3.6 

18 

718997 

722999 

726965 

730896 

734791 

73865i 

4i 

391.3- 

J9 

742478 

746270 

75oo3i 

753758 

757455 

761119 

4o 

371.2 

20 

764754 

768358 

771932 

775477 

778904 

782482 

39 

353.1 

21 

785943 

789376 

792782 

796162 

7995i5 

802843 

38 

336.7 

22 

806146 

809423 

812677 

815906 

819111 

822292 

37 

321.7 

23 

82545i 

828586 

831700 

834791 

837860 

840907 

36 

3o8.o 

24 

843934 

846939 

849924 

852889 

855833 

858757 

35 

295.4 

25 

861662 

864548 

8674i5 

870262 

873092 

8-75902 

34 

283.8 

26 

878695 

881470 

884228 

886968 

889690 

892396 

33 

273.1 

27 

895085 

897758 

900414 

9o3o54 

905678 

908287 

32 

263.2 

28 

910879 

913457 

916019 

9i8566 

921098 

923616 

3i 

254.0 

29 

926119 

928608 

931082 

933543 

935989 

938422 

3o 

245.4 

3o 

94o84a 

943248 

94564i 

948020 

950387 

952741 

29 

237.3 

3i 

955082 

957411 

959727 

962031 

964322 

966602 

28 

229.8 

32 

968870 

971126 

97337o 

9756o3 

977824 

980034 

27 

222.7 

33 

982233 

984421 

986598 

988764 

990919 

993064 

26 

2:6.1 

34 

995198 

997322 

999435 

*8.ooi538 

S.oo363i 

8.005714 

25 

209.8 

35 

8.007787 

S.oogSSo 

8.011903 

013947 

015981 

oi8oo5 

24 

203.9 

36 

O2OO2I 

022027 

024023 

026011 

027989 

029959 

23 

198.3 

37 

©31919 

033871 

o358i4 

037749 

039675 

041592 

22 

IQ3.0 

38 

o435oi 

o454oi 

047294 

049178 

o5io54 

©52922 

21 

188.0 

39 

054781 

o56633 

o58477 

o6o3i4 

062142 

o63963 

2O 

i83.2 

4o 

065776 

067582 

069380 

071171 

072955 

o7473i 

J9 

178.7 

4i 

076500 

078261 

080016 

081764 

o835o4 

o85238 

18 

174.4 

42 

086965 

088684 

090398 

092104 

093804 

095497 

I7 

170.3 

43 

097183 

098863 

ioo537 

IO22O4 

io3864 

io55i9 

16 

166.4 

44 

107167 

108809 

no444 

II2O74 

113697 

n53i5 

i5 

162.6 

45 

116926 

ii8532 

I2Ol3l 

I2I725 

I233i3 

124895 

i4 

i59.i 

46 

126471 

128042 

129607 

i3:r66 

132720 

134268 

i3 

i55.6 

47 

i358io 

i37348 

138879 

140406 

141927 

143443 

12 

i52.4 

48 

144953 

146458 

147959 

149453 

i  50943 

152428 

II 

149.2 

49 

153907 

i55382 

i56852 

i583i6 

i59776 

i6i23i 

10 

146.2 

5o 

162681 

164126 

165566 

167002 

168433 

169859 

9 

i43.3 

5i 

171280 

172697 

174109 

175517 

176920 

178319 

8 

i4o.5 

52 

179713 

i8no3 

182488 

183869 

i85245 

186617 

7 

i37-9 

53 

187985 

189348 

190707 

192062 

I934i3 

194760 

6 

i35.3 

54 

196102 

197440 

198774 

2OOIO4 

2oi43o 

202752 

K 

i32.8 

55 

204070 

2o5384 

206694 

2O8OOO 

209302 

210601 

4 

i3o.4 

56 

211895 

2i3i85 

214472 

2i5755 

217034 

218309 

3 

128.1 

57 

219681 

220849 

222II3 

223374 

22463i 

225884 

2 

125.9 

58 

227134 

22838o 

229622 

23o86i 

232096 

233328 

I 

123.7 

59 

234557 

235782 

237003 

238221 

239436 

24o647 

O 

121.  6 

60" 

50" 

40" 

30"        20" 

10" 

Co-sine  of  59  Degrees. 

Min. 

LOGARITHMIC    TANGENTS. 


Mm. 

Tangent  of  0  Degree. 

P.  Par 

*T»  1  • 

0" 

10" 

20" 

30" 

40"    |    50" 

to  j.  . 

0 

Inf.  Neg. 

5.685575 

5.  9866o5 

6.162696 

6.287635 

6.384545 

59 

I 

6.463726 

6.53o673 

6.588665 

639817 

685575 

726968 

58 

2 

764?56 

7995i8 

83i7o3 

861666 

889695 

916024 

57 

3 

94o84y 

964329 

986605 

7.007794 

7  027998 

7.o473o3 

56 

4 

7.065786 

7.o835i5 

7.ioo548 

116939 

i32733 

i47973 

55 

5 

162696 

i76937 

190725 

204089 

217054 

229643 

54 

6 

241878 

253777 

265359 

276640 

287635 

298359 

53 

7 

3o8825 

319044 

329028 

338788  '   348333 

357673 

52 

8 

366817 

375772 

384546 

393i46 

401579 

409852 

5i 

9 

417970 

425939 

433764 

44i/5i 

449004 

456428 

5o 

10 

463727 

470906 

477968 

4*4917 

491756 

498490 

49 

689.; 

ii 

5o5i2o 

5n65i 

5i8o85 

^24426 

530675 

536835 

48 

629.  L 

12 

542909 

548900 

5548o8 

!'6o638    56639o 

572o68 

47 

679.] 

i3 

577672 

583204 

588667 

594062    599391 

6o4655 

46 

536.  s 

U 

609857 

614996 

620076 

625097 

63oo6o 

634968 

45 

499-- 

i5 

639820 

644619 

649366 

654o6i 

658706 

663302 

44 

467.< 

16 

667849 

672350 

676804 

68i2i3 

685578 

689900 

43 

438.^ 

17 

694179 

698416 

•702612 

706768 

7io885 

7i4963 

42 

4i3.( 

18 

719003 

723oo5 

726972 

730902 

734797 

738658 

4i 

39i.: 

'9 

742484 

746277 

750037 

753765 

757462 

761127 

4o 

371.. 

20 

764761 

768365 

771940 

775485 

779002 

782490 

39 

353.1 

21 

78595i 

789384 

792790 

796170 

799524 

802852 

38 

336.-; 

22 

8o6i55 

809433 

812686 

8i59i5 

819120 

822302 

37 

321.-; 

23 

82546o 

828596 

831710 

8348oi 

837870 

840918 

36 

3o8.c 

24 

843944 

846950 

849935 

852900 

855844 

858769 

35 

295.2 

25 

861674 

86456o 

867426 

870274 

873104 

8759i5 

34 

283.  c 

26 

878708 

88i483 

884240 

•  886981 

889704 

892410 

33 

273.5 

27 

895099 

897772 

900428 

903068 

905692 

908301 

32 

263.  i 

28 

910894 

9i347i 

9i6o34 

9i858i 

921113 

92363! 

3i 

254.  c 

29 

926134 

928623 

931098 

933559 

936006 

938439 

3o 

245.2 

3o 

94o858 

943265 

945658 

948o37 

95o4o4 

952758 

29 

237.: 

3i 

955ioo 

957428 

959745 

962049 

964341 

966621 

28 

229.  £ 

32 

968889 

97n45 

973389 

975622 

977844 

980054 

27 

222.  •; 

33 

982253 

98444i 

986618 

988785 

99o94o 

993o85 

26 

2l6.i 

34 

995219 

997343 

999457 

8.ooi56o 

8.oo3653 

3.oo5736 

25 

2O9.^ 

35 

8  007809 

8.oo9872 

8.011926 

013970 

016004 

018029 

24 

203  C 

36 

020044 

O22o5l 

024048 

026o35 

028014 

'  029984 

23 

i98J 

3? 

031945 

033897 

o3584o 

o37775 

039701 

041618 

22 

I93.c 

38 

043527 

045428 

047321 

049205 

o5io8i 

©52949 

21 

i88.c 

39 

054809 

o56662 

o585o6 

060342 

062171 

063992 

20 

i83.2 

4o 

o658o6 

067612 

069410 

O7I2OI 

072985 

074761 

T9 

178.7 

4i 

076531 

078293 

o8oo47 

081795 

o83536 

085270 

18 

174.4 

42 

086997 

088717 

090431 

092137 

093837 

ogSSSo 

17 

170.2 

43 

097217 

098897 

100571 

102239 

loSgoo 

io5554 

16 

166.4 

44 

107203 

io8845 

110481 

II2IIO 

n3734 

n5352 

i5 

162.  7 

45 

116963 

118569 

120169 

I2i763 

i2335i 

124933 

i4 

i59.  i 

46 

i265io 

128081 

129646 

i3i2o6 

132760 

i343o8 

i3 

I55.-J 

47 

i3585i 

i37389 

138921 

140447 

141969 

i43485 

12 

162.4 

48 

144996 

i465oi 

i48ooi 

149497 

150987 

152472 

II 

149-3 

49 

i53952 

i55426 

156896 

i5836i 

159821 

161276 

10 

146.2 

5o 

162727 

164172 

i656i3 

167049 

168480 

169906 

9 

143.4 

5i 

171328 

172745 

I74i58 

i75566 

176969 

178368 

8 

i4o.C 

52 

i79763 

i8n53 

182538 

183919 

i85296 

186668 

7 

187  v 

53 

i88o36 

189400 

190760 

192115 

193466 

194813 

6 

135.2 

54 

196156 

197494 

198829 

200159 

2oi485 

202808 

5 

I32.S 

55 

204126 

2o544o 

206750 

208057 

209359 

2io658 

4 

i3o.^ 

56 

211953 

213243 

2i453o 

2i58i4 

217093 

218369 

3 

128.1 

57 

219641 

220909 

222174 

223434 

224692 

225945 

2 

125.  <; 

58 

227195 

228442 

229685 

230924 

232l6o 

233392 

I 

I23.fi 

59 

234621 

235846 

237068 

238286 

339502 

240713 

O 

121  .'J 

60"        50"        40" 

30" 

20" 

10' 

Co-tangent  of  89  Degrees. 

LOGA  KITH  MIC     f3  I  ft  £  3. 


Min 

Sine  of  1  Degree. 

P.  Part 

O7 

10" 

20" 

30"        40" 

50" 

to  I". 

0 

8.24i855 

8.  243060 

8.244261 

8.245459  8.246654 

8.247845- 

69 

n9.6 

I 

249o33 

260218 

25l4oO 

262678   263763 

264926 

58 

117  7 

2 

266094 

267260 

258423 

a59582    26o739 

261892 

57 

n5.3 

3 

263o4a 

264i9o 

265334 

266476   267613 

268749 

56 

n4.o 

4 

26988i 

27IOIO 

272137 

273260   274381 

276499 

55 

112.  2 

5 

276614 

277726 

278835 

27994i  :   281046 

282145 

54 

1*0.5 

6 

283243 

284339 

28543i 

286621    287608 

288692 

53 

108.8 

7 

289773 

29o852 

29I928 

293oo2    294o73 

296141 

52 

107.2 

8 

2962O7 

297270 

29833o 

299388    3oo443 

301496 

5i 

106.7 

9 

302546 

3o3594 

3o4639 

3o568i    306721 

3o7759 

5o 

104.1 

10 

8.3o8794 

8.3o9827 

8.310867 

8.3n885 

8.3I29IO 

8.3i3933 

49 

102.6 

ii 

3i4954 

3i5972 

316987 

3i8ooi 

3I9OI2 

320O2I 

48 

IOI  .2 

12 

321027 

322031 

323o33 

324032 

325o29 

326024 

47 

99.8 

i3 

327016 

328oo7 

328996 

32998o   33o964 

33i945 

46 

98.5 

i4 

332924 

333901 

334876 

335848   3368i9 

337787 

45 

97.1 

16 

338753 

339717 

340679 

34i638 

342696 

34355i 

44 

96.8 

16 

3445o4 

345456 

3464o5 

347352 

348297 

349240 

43 

94.6 

17 

35oi8i 

35.1119 

362066 

35299i    353924 

354855 

42 

93.4 

18 

355783 

3567io 

357635 

358558   359479 

360398 

4i 

92.2 

J9 

36i3i5 

36223o 

363i43 

364o55  '   364964 

36587i 

4o 

91  .0 

20 

8.  366777 

8.36768i 

8.368582 

8.369482  8.37o38o 

8.371277 

.  39 

89.9 

21 

372171 

373o63 

373954 

374843   376730 

3766i5 

38 

88.8 

22 

377499 

37838o 

379260 

38oi38   38ioi5 

881889 

37 

87.7 

23 

382762 

383633 

384602 

38537o   386236 

387ioo 

36 

86.7 

24 

3879162 

388823 

389682 

390639   391396 

392249 

35 

85.6 

25 

393ioi 

393961 

394800 

396647   396493 

397337 

34 

84.6 

26 

398i79 

399020 

399869 

400696   4oi532 

402366 

33 

83.7 

27 

4o3i99 

4o4o3o 

404869 

4o5687 

4o65i4 

4o7338 

32 

82.7 

28 

408161 

408983 

409803 

410621 

4n438 

412264 

3i 

81.8 

29 

4i3o68 

4i388o 

414691 

4i55oo 

4i63o8 

417114 

3o 

80.8 

3o 

8.4i79i9 

8.418722 

8.419624 

8.420326 

8.42II23 

8.421921 

29 

80.0 

3i 

422717 

4235n 

*4243o4 

426096 

426886 

426675 

28 

79.1 

32 

427462 

428248 

429032 

429816 

430697 

43i377 

27 

78.2 

33 

432i56 

432934 

433710 

434484   436267 

436029 

26 

77-4 

34 

4368oo 

437569 

438337 

439io3   439868 

44o632 

25 

76.6 

35 

44i394 

442166 

44s9i5 

443674   44443i 

445i86 

24 

76.8 

36 

44594i 

446694 

447446 

448196   448946 

449694 

23 

76.0 

37 

45o44o 

45u86 

45i93o 

462673   4534i4 

454i54 

22 

74.2 

38 

454893 

45563i 

456368 

457io3  '   467837 

45857o 

21 

73.5 

39 

4593oi 

46oo32 

460761 

461489 

462216 

462941 

2O 

72-7 

4o 

8.463665 

8.464388 

8.466110 

8.465830 

8.466660 

8.467268 

I9 

72.O 

4i 

467985 

4687oi 

4694i6 

470129 

470841 

47i553 

18 

71.3 

42 

472263 

472971 

473679 

474386 

476091 

475795 

17 

70.6 

43 

476498 

477200 

4779oi 

478601 

479299 

479997 

16 

69.9 

44 

48o693 

48i388 

482083 

482776 

483467 

484i58 

i5 

69.2 

45 

484848 

485536 

486224 

486910 

487696 

488280 

i4 

68.6 

46 

488963 

489645 

49o326 

491006 

491686 

492363 

i3 

67.9 

47 

493o4o 

493716 

49439o 

496064 

496736 

496408 

12 

67.3 

48 

497o78 

497748 

4984i6 

499084   499760 

5oo4i6 

II 

66.7 

49 

601080 

5oi743 

602406 

603067 

603727 

5o4386 

IO 

66.1 

5o 

8.  5o5o45 

8.606702 

8.5o6358 

8.607014 

8.607668 

8.5o832i 

9 

65.5 

61 

5o8974 

609626 

610276 

610926 

5ii573 

612221 

8 

64-9 

62 

5i2867 

5i35i3 

614167 

5i48oi 

5i5444 

616086 

7 

64.3 

53 

5i6726 

5i7366 

618006 

5i8643 

619280 

619916 

6 

63.7 

54 

620661 

621186 

52i8i9 

622461 

523o83 

5237i3 

5 

63.2 

55 

524343 

624972 

525599 

626226 

626862 

527477 

4 

62.6 

56 

628102 

628726 

529347 

629969 

530690 

531209 

3 

62.1 

07 

53i828 

532446 

533o63 

533679 

534296 

534909 

2 

61.6 

58 

535523 

536i36 

536747 

537358 

537969 

538578 

I 

61.1 

59 

539i86 

539794 

641007 

541612 

642216 

O 

60.6 

60" 

50"        40" 

30"    '    20" 

10" 

1\  tlw. 

Co-sine  of  88  Degrees. 

.LOGARITHMIC    TANGENT ». 


\  Min                Tangent  of  1  Degree. 

P.  Part 

0" 

10" 

20" 

30" 

40"        50" 

to  1". 

0 

8.241921 

8.243126 

8.244328 

8.245526 

8.246721 

8.247913 

59 

119.7 

i 

249102 

250287 

261469 

252648 

253823 

264996 

58 

117.7 

2 

256i65 

25733i 

258494 

269664 

260811 

261966 

57 

116.8 

3 

263n5 

264263 

266408 

266649 

267688 

268824 

56 

n4«o 

4 

269956 

271086 

272213 

273337 

274458 

276676 

55 

112.  2 

5 

276691 

277804 

278913 

280020 

281124 

282226 

54 

no.  5 

6 

283323 

284419 

286612 

286602 

287689 

288774 

53 

108.9 

7 

289856 

290935 

292012 

293086 

294167 

296226 

62 

107.2 

8 

296292 

297355 

298416 

299474 

3oo53o 

3oi583 

61 

106.7 

9 

302634 

3o3682 

304727 

306770 

3o68ii 

307849 

5o 

104.2 

10 

8.3o8884 

8.309917 

8.310948 

8.311976 

8.3i3oo2 

8.3i4o25 

49 

IO2.7 

ii 

3i5o46 

3i6o65 

317081 

318096 

319106 

320116 

48 

101.3 

12 

32II22 

322127 

323129 

324129 

326126 

326121 

47 

99.8 

i3 

327II4 

328io5 

329093 

33oo8o 

33io64 

332045 

46 

98.5 

i4 

333025 

334oo2 

334977 

33595o 

336921 

337890 

45 

97.2 

i5 

338856 

339821 

340783 

341743 

342701 

343657 

44 

96.9 

16 

3446io 

345562 

3465i2 

347459 

3484o5 

349348 

43 

94.6 

J7 

350289 

351229 

362166 

353ioi 

354o35 

354966 

42 

93.4 

18 

355895 

356823 

357748 

368671 

359593 

36o5i2 

4i 

92.2 

J9 

36i43o 

362345 

363269 

364171 

365o8i 

365988 

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91.1 

20 

8.366895 

8.367799 

8.368701 

8.369601 

8.370600 

8.37i397 

39 

89.9 

21 

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373i85 

374076 

374966 

375853 

376738 

38 

88.8 

22 

377622 

3785o4 

379385 

380263 

38ii4o 

382016 

37 

87.8 

23 

382889 

38376o 

38463o 

386498 

386364 

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86.7 

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388092 

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3898i3 

390670 

391626 

39238i 

35 

85.7 

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396782 

396628 

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83.7 

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82.8 

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4o83o4 

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81.8 

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4i32i3 

414026 

4i4837 

416647 

4i6456 

417263 

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80.9 

3o 

8.4i8o68 

8.418872 

8.419674 

8.420476 

8.421274 

8.422072 

29 

80.0 

3i 

422869 

423664 

424468 

426260 

426041 

42683o 

28 

79.1 

32 

427618 

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429189 

429973 

430766 

43i536 

27 

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73.5 

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8.463849 

8.464572 

8.466296 

8.  466016 

8.466736 

8.467455 

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47io3i 

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474579 

476286 

476990 

17 

70.7 

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476693 

477396 

478097 

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70.0 

44 

480892 

48i588 

482283 

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69.3 

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48574o 

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487116 

487801 

488486 

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68.6 

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489170 

489852 

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491216 

491894 

492673 

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47 

49325o 

493927 

494602 

496276 

496949 

496622 

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497293 

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498632 

499300 

499967 

5oo633 

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66.8 

49 

501298 

501962 

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5o3948 

5o46o8 

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8.  505267 

8.  506925 

8.  606682 

8.607238 

8.607893 

8.608647 

9 

65.5 

5i 

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5io5o3 

5in53 

611802 

612461 

8 

65.o 

52 

SiSogS 

5i3744 

5i4389 

5i5o34 

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5i632o 

7 

64.4 

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5i696i 

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618880 

619618 

620164 

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63.8 

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520790 

621426 

622069 

622692 

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5 

63.3 

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62.7 

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53i46o 

3 

62.2 

57 

532o8o 

532698 

5333i6 

533933 

534549 

535i64 

2 

61.6 

58 

535779   536392 

537006 

537617 

538227 

538837 

I 

61.1 

59 

539447   54oo55 

64^662 

541269 

541876 

642480 

O 

60.6 

60"        50" 

40" 

30"        20" 

10" 

Co-tangent  of  88  Degrees. 

*  m  .  , 

Min. 

LOGARITHM    c    SINES. 


d 

Sine  of  2  Degrees. 

a 

Sine  of  3  Degrees. 

~~~ 

m 

0'' 

10* 

20" 

30-'' 

40" 

50" 

3 

0"    |  10-'  |  20" 

30" 

40'' 

50" 

c 

8.542819 

3422 

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462^ 

522/ 

582 

59 

c 

8.718800 

9202 

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i 

6422 

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8212 

880^ 

94o 

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2002 

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3i97 

58 

2 

9995 

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11  75 

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236i 

295o 

57 

2 

3595 

3992 

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4785 

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57 

|    5 

8  553539 

4126 

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5885 

647o 

56 

J 

5972 

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6762 

7i56 

755o 

/  / 

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56 

i 

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8801 

938i 

996 

55 

L 

8337 

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9906 

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55 

t 

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t 

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54 

6 

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7 

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8000 

8569 

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7 

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8 

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S787 

7I9° 

7593 

7996 

8398 

0 

59 

8.841774 

2076 

2378 

2680 

2982 

3283 

0 

60" 

50" 

40" 

30" 

20" 

10" 

CJ 

6U" 

50" 

40" 

30" 

20" 

10" 

C 

Co-sine  of  87  Degrees. 

s 

Co-sine  of  86  Degrees. 

a 

pp  Al"V'  3"  4"  5"  6"  7"  8"  9" 

„  v  ,  (  I"  2"  3"  4"  5"  6"  7"  8"  9" 

iri\  48  96  145  19?  241  289  338  386  434 

r.^art^  34  69  1Q3  13g  172  2Q7  241  275  31() 

LOGARITHMIC    TANGENT* 


27 


.2 

Tangent  of  2  Degrees. 

.S 

Tangent  of  3  Degrees. 

s 

0" 

10"   20" 

30"  |  40 

50" 

8 

0" 

10" 

20" 

30" 

40"  |  50" 

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8.719396 

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54 

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6726 

5 

54 

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4234 

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4853 

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5 

55 

7140 

7554 

7967 

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8794 

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4 

55 

547i 

6780 

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6397 

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4 

56 

9618 

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3 

56 

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7629 

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8867 

3 

57 

8.712083 

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2902 

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4127 

2 

57 

9i63 

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9776 

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.692 

2 

58 

4534 

494s 

5348 

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6161 

6567 

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58 

8.84o998 

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1607 

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2216 

2621 

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59 

6972 

7377 

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59 

2826 

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3735 

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0 

60" 

50"   40" 

30" 

20" 

10" 

60" 

50" 

40" 

30" 

20" 

10"  I 

Co-tangent  of  87  Degrees. 

£ 

Co-tangent  of  86  Degrees.   |  § 

P  P  A  1"  2"  3"  4"  5"  6"  7"  8"  9" 

p  p   <  1"  2"  3"  4"  5"  6"  7"  8"  !)" 

iri\  48  97  145  193  242  290  338  387  435 

iTll  35  69  104  138  173  207  242  276  311 

LOGAKITHMIC      SlNES. 


1 

Sine  of  4  Degrees. 

j 

Sine  of  5  Degrees. 

s 

0' 

10" 

20" 

30" 

40" 

50" 

9 

0" 

10" 

20" 

30" 

40" 

50" 

o 

8.843585 

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4487 

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59 

o 

8.94o296 

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58 

i 

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2696 

2935 

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2 

7i83 

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8078 

8376 

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57 

2 

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3 

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3 

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4 

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5 

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5 

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6 

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5212 

5456 

5699 

3 

56 

44oo 

4602 

48o5 

5oo7 

5209 

54n 

3 

57 

5942 

6i85 

6428 

6671 

6914 

7i56 

2 

57 

56i3 

58i5 

60  1  7 

6219 

6421 

6622 

2 

58 

7398 

7641 

7883 

8i25 

8366 

8608 

I 

58 

6824 

7025 

7227 

742£ 

762C 

783o 

I 

59 

885o 

9o9i 

9332 

9573 

9814 

..55 

0 

59 

8o3i 

8232 

8433 

8633 

8834 

9o34 

0 

60" 

50" 

40" 

30" 

20" 

10" 

. 

60" 

50"  |  40"  |  30"  |  20" 

io" 

g 

Co-sine  of  85  Degrees. 

a 

Co-sine  of  84  Degrees. 

.<  1"2"3"  4"  5"  6"  7"  8"  9" 
"  I  27  53  80  107  134  160  187  214  241 

P  Port  5  *"  ~"  3"  4//   5"   6"   ?//   8"   °" 

r>rart)22  44  66  87  109  131  153  175  197 

.  -,  ..  .  _-  -  - 

LOGARITHMIC    TANGENTS 


1 

Tangent  of  4  Degrees. 

.5 

Tangent  of  5  Degrees. 

s 

V 

10" 

20" 

30" 

40" 

50" 

3| 

0" 

10" 

20" 

30"  |  40" 

50" 

o 

8.844644 

4946 

5248 

555i 

5852 

6i54 

>9 

o 

8.941952 

2i94 

2437 

2670 

2921 

3i63 

59 

I 

6455 

6757 

7o58 

7358 

765q 

7959 

58 

i 

34o4 

3646 

3888 

4i29 

4370 

46ii 

58 

2 

8260 

856o 

8859 

9l59 

9458 

9758 

57 

2 

4852 

5o93 

5334 

5574 

58i5 

6o55 

57 

3 

S.SSooSy 

o355 

o654 

0952 

1260 

1  548 

56 

3 

6295 

6535 

6775 

7015 

7255 

7494 

56 

4 

i846 

2i44 

244i 

2738 

3o35 

3332 

55 

4 

7734 

7973 

8212 

845  1 

8690 

8929 

55 

5 

3628 

3925 

4221 

45i7 

48i3 

5io8 

54 

5 

9168 

9406 

9644 

9883 

.121 

.359 

54 

6 

54o3 

5699 

5993 

6288 

6583 

6877 

53 

6 

8.950597 

o834 

I072 

i3o9 

1  547 

1784 

53 

7 

7171 

7465 

7759 

8o53 

8346 

8639 

52 

7 

2021 

2258 

2495 

2732 

2968 

32o5 

52 

8 

8932 

9225 

95l7 

9810 

.  IO2 

•  394 

5i 

8 

344i 

3677 

3913 

4i49 

4385 

4621 

5i 

9 

8.860686 

0977 

1269 

i56o 

i85i 

2142 

5o 

9 

4856 

5092 

5327 

5562 

5797 

6o32 

5o 

10 

2433 

2723 

3oi3 

33o3 

3593 

3883 

49 

IO 

6267 

65o2 

6736 

6971 

72o5 

7439  4o 

ii 

4i73 

4462 

475i 

5o4o 

5329 

56i7 

48 

ii 

7674 

7908 

8i4i 

8375 

8609 

8842 

48 

12 

59o6 

6194 

6482 

6769 

7°57 

7344 

47 

12 

9o75 

93o9 

9542 

9775 

...8 

.240 

47 

i3 

7632 

7919 

8206 

8492 

8779 

9o65 

46 

i3 

8.0.60473 

0705 

0938 

1170 

l4o2 

i634 

46 

i4 

935i 

9637 

9923 

.208 

.494 

•779 

45 

i4 

1866 

2098 

2329 

256i 

2792 

3o23 

45 

i5 

8,871064 

1  349 

i633 

1918 

22O2 

2486 

44 

i5 

3255 

3486 

37I6 

3947 

4178 

44o8 

44 

16 

2770 

3o54 

3337 

3620 

3904 

4i87 

43 

16 

4639 

4869 

5o99 

5329 

5559 

5789 

43 

17 

4469 

4752 

5o34 

53i7 

5599 

588i 

42 

17 

6oi9 

6248 

6478 

6707 

6g36 

7i65 

42 

18 

6162 

6444 

6725 

7006 

7287 

7568 

4i 

18 

7394 

7623 

7852 

8081 

8309 

8538 

4i 

J9 

7849 

8129 

8409 

8689 

8969 

9249 

4o 

'9 

8766 

8994 

9222 

945o 

9678 

99o5 

4o 

20 

9529 

9808 

..87 

.366 

.645 

.924 

39 

20 

8.970133 

o36o 

o588 

o8i5 

IO42 

1269 

39 

21 

a.  881202 

i48o 

1759 

2037 

23i4 

2592 

38 

21 

1496 

1723 

1949 

2176 

24O2 

2628 

38 

22 

2869 

3i47 

3424 

3701 

3977 

4254 

37 

22 

2855 

3o8i 

33o7 

3532 

3758 

3984 

37 

23 

453o 

4807 

5o83 

5358 

5634 

5910 

36 

23 

4209 

4435 

466o 

4885 

5uo 

5335 

36 

24 

6i85 

646o 

6735 

7010 

7285 

7559 

35 

24 

556o 

5784 

6009 

6233 

6458 

6682 

35 

25 

7833 

8108 

8382 

8655 

8929 

9202 

34 

25 

6906 

7i3o 

7354 

7578 

78oi 

8025 

34 

26 

9476 

9749 

.  .22 

.295 

.567 

.84o 

33 

26 

8248 

8472 

8695 

89i8 

9i4i 

9364 

33 

27 

8  891112 

i384 

i656 

I928 

2199 

2471 

32 

27 

9586 

98o9 

..32 

.254 

.476 

.699 

32 

28 

2742 

3oi3 

3z84 

3555 

3825 

4o96 

3i 

28 

8.980921 

n43 

i364 

i586 

!8o8 

2029 

3i 

29 

4366 

4636 

4906 

5176 

5445 

57i5 

3o 

29 

225l 

2472 

2693 

29i4 

3i35 

3356 

3o 

3o 

5984 

6253 

6522 

6791 

7060 

7328 

29 

3o 

3577 

3798 

4oi8 

4238 

4459 

4679 

29 

3i 

7596 

7864 

8182 

84oo 

8668 

8935 

28 

3i 

4899 

5n9 

5339 

5559 

5778 

5998 

28 

32 

9203 

9470 

9737 

,..4 

.270 

.537 

27 

32 

6217 

6437 

6656 

6875 

7o94 

73i3 

27 

33 

8.9oo8o3 

io69 

i335 

1601 

1867 

2l32 

26 

33 

7532 

775o 

7969 

8i87 

84o6 

8624 

26 

34 

2398 

2663 

2928 

3193 

3458 

3722 

25 

34 

8842 

9o6o 

9278 

9496 

97i4 

993i 

25 

35 

3987 

425i 

45i5 

4779 

5o43 

53o6 

24 

35 

8.900149 

o366 

o583 

0801 

1018 

1235 

24 

36 

5570 

5833 

6o96 

6359 

6622 

6885 

23 

36 

i45i 

1668 

i885 

2101 

23  1  8 

2534 

23 

37 

7147 

74io 

•7672 

7934 

8196 

8457 

22 

37 

275o 

2966 

3i82 

3398 

36i4 

383o 

22 

38 

8710 

898o 

9242 

95o3 

9764 

..25 

21 

38 

4o45 

4261 

4476 

4692 

4907 

5l22 

21 

39 

8.910285 

o546 

0806 

1066 

i326 

i586 

2O 

39 

5337 

5552 

5766 

598i 

6196 

64io 

20 

4o 

1  846 

2106 

2365 

2624 

2883 

3l42 

19 

4o 

6624 

6839 

7o53 

726? 

748  1 

7694 

I9 

4i 

34oi 

366o 

39i8 

4i77 

4435 

4693 

18 

4i 

79o8 

8122 

8335 

8549 

8762 

8975 

18 

42 

495i 

52O9 

5466 

5724 

598i 

6238 

17 

42 

9i88 

94oi 

96i4 

982? 

..4o 

.252 

I7 

43 

64g5 

6752 

70o9 

7265 

7522 

7778 

16 

43 

9.ooo465 

o677 

o889 

IIO2 

i3*t4 

i526 

16 

44 

8o34 

829.0 

8546 

8801 

9o57 

93l2 

i5 

44 

i738 

i949 

2161 

2373 

2584 

2795 

i5 

45 

9568 

9823 

..78 

.332 

.587 

.84i 

i4 

45 

8007 

32i8 

3429 

364o 

385i 

4o6i 

i4 

46 

8.92io96 

i35o 

1604 

i858 

2112 

2365 

i3 

46 

4272 

4483 

4693 

49o4 

5n4 

5324 

i3 

47 

26l9 

2872 

3i25 

3378 

363i 

3884 

12 

47 

5534 

5744 

5954 

6164 

6373 

6583 

12 

48 

4i36 

4389 

464i 

4893|5i45 

5397 

II 

48 

670.2 

7OO2 

721.1 

7420 

^7629 

7838 

II 

49 

5649 

59oo 

6i52 

64o3 

6654 

69o5 

IO 

49 

8047 

8256 

8465 

8673 

8882 

9o9o 

IO 

5o 

7i56 

7407 

7657 

7908 

8i58 

84o8 

9 

5o 

9298 

95o7 

97i5 

9923 

.i3i 

.338 

9 

5i 

8658 

89o8 

9i58 

9407 

9657 

99o6 

8 

5i 

9.oio546 

o754 

0961 

n69 

i376 

i583 

8 

52 

8.93oi55 

o4o4 

o653 

0902 

n5o 

!399 

7 

52 

1790 

i997 

2202; 

2411 

2618 

2824 

7 

53 

1  647 

i895 

2143 

2891 

2639 

2887 

6 

53 

3o3i 

3237 

3444 

365o 

3856 

4062 

6 

54 

3i34 

338i 

3629 

3876 

4i23 

4369 

5 

54 

4268 

4474 

468o 

4886 

Sogi 

5297 

5 

55 

46i6 

4862 

5io9 

5355 

56OI 

5847 

'4 

55 

55o2 

57o7 

59i3 

6118 

6323 

6528 

4 

56 

6o93 

6339 

6584 

683o 

7075 

7320 

3 

56 

6732 

6937 

7142 

7346 

755i 

7755 

3 

57 

7565 

7810 

8o55 

8299 

8544 

8788 

2 

57 

7959 

8i64 

8368 

8572 

8776 

8979 

2 

58 

9o32 

9276 

9520 

9764 

,25l 

I 

58 

9183 

9387 

9590 

9794 

9997 

.200 

I 

59 

8.94o49^ 

o738 

o98i 

122^ 

1467 

1709 

O 

59 

9.O2o4o? 

0606 

0809 

1012 

I2l5 

i4i8 

O 

60" 

50" 

40" 

30" 

20" 

10" 

. 

60" 

50" 

40" 

30*  I  20" 

10" 

. 

Co-tangent  of  85  Degrees. 

a 

Co-tangent  of  84  Degrees. 

1 

p  Pftrt$  1"  2"  3"  4"  5"  6"  7"  8"  9" 
I  27  54  81  108  135  162  188  215  242 

p  pirtU"  2"  3"  4"  5"  6"  7"  8"  9" 
ml  22  44  66  88  110  132  154  177  199 

LOGARITHMIC    SINES. 


Sine  of  6  Degrees. 

1 

Sine  of  7  Degrees. 

s 

0" 

10" 

20" 

30" 

40" 

w 

s 

0" 

10" 

20" 

F30" 

40""1 

50" 

0 

9,oi9235 

9435 

9635 

9835 

.  .35 

.235 

59 

o 

9.o85894 

6066 

6237 

64o9 

658o 

675i 

59 

I 

9.020435 

o635 

o834 

io34 

1233 

i433 

58 

I 

6922 

7093 

7264 

7435 

7606 

7777 

58 

2 

:632 

i83i 

2o3o 

2229 

2428 

2627 

57 

2 

7947 

8118 

8288 

8459 

8629 

8800 

57 

3 

2825 

3o24 

3223 

3421 

36i9 

38i8 

56 

3 

897o 

9i4o 

93io 

948o 

965i 

982O 

56 

4 

4oi6 

42i4 

44i2 

46  1  o 

48o7 

5oo5 

55 

4 

9990 

.160 

.33o 

.5oo 

.669 

.839 

55 

5 

52o3 

54oo 

5598 

5795 

5992 

6189 

54 

5 

9.091008 

1178 

i347 

i5i6 

i685 

i855 

54 

6 

6386 

6583 

6780 

6977 

7174 

737o 

53 

6 

2024 

2I93 

2362 

253o 

2699 

2868 

53 

7 

7567 

7763 

7960 

8i56 

8352 

8548 

52 

7 

3o37 

32o5 

3374 

3542 

37n 

3879 

52 

8 

8744 

8940 

9i36 

9332 

9527 

9723 

5i 

8 

4047 

4216 

4384 

4552 

4720 

4888 

5i 

9 

99i8 

.114 

.309 

.5o4 

•699 

5o 

9 

5o56 

5223 

539i 

5559 

5726 

5894 

5o 

10 

9.o3io89 

1284 

1479 

i673 

186? 

2062 

49 

10 

6062 

6229 

6396 

6564 

673i 

6898 

49 

ii 

2257 

245i 

2645 

2839 

3o33 

3227 

48 

ii 

7065 

7232 

7399 

7566 

7733 

79oo 

48 

12 

342i 

36i5 

38o9 

40O2 

4196 

4389 

47 

12 

8066 

8233 

8399 

8566 

8732 

8899 

4? 

i3 

4582 

4776 

4969  5i62 

5355 

5548 

46 

i3 

9065 

923l 

9398 

9564 

973o 

9896 

46 

r4 

574i 

5933 

6126  6819 

65n 

67o3 

45 

i4 

9.  100062 

0227 

o393 

o559 

0725 

o89o 

45 

i5 

6896 

7088 

7280 

7472 

7664 

7856 

44 

iS 

io56 

1221 

i387 

i552 

1717 

i883 

44 

16 

8o48 

8239 

843i 

8623 

8814 

9oo5 

43 

16 

2048 

2213 

2378 

2543 

2708 

2873 

43 

I7 

9i97 

9388 

9579 

977° 

996i 

.152 

42 

17 

3o37 

32O2 

3367 

353i 

3696 

386o 

42 

18 

9.o4o342 

o533 

0724 

o9i4 

no5 

I295 

4i 

1.8 

4025 

4i89 

4353 

45i7 

4682 

4846 

4i 

i9 

i485 

i675 

i865 

2o55 

2245 

2435 

4o 

i9 

5oio 

5i74 

5337 

55oi 

5665 

5829 

4o 

20 

2625 

28!5 

3oo4 

3i94 

3383 

3572 

39 

20 

5992 

6i56 

63i9 

6483 

6646 

6810 

39 

21 

3762 

395i 

4i4o 

4329 

45i8 

47o7 

38 

21 

6973 

7i36 

7299 

7462 

7625 

7788 

38 

22 

4895 

5o84 

5273 

546i 

565o 

5838 

37 

22 

795i 

8n4 

82?7 

8439 

8602 

8765 

37 

23 

6026 

6214 

6402 

659o 

6778 

6966 

36 

23 

8927 

9o9o 

9252 

94i4 

9577 

9739 

36 

24 

7i54 

7342 

7529 

7717 

7904 

8o9i 

35 

24 

9901 

..63 

.225 

.387 

•  549 

.711 

35 

25 

8279 

8466 

8653 

884o 

9027 

92l4 

34 

25 

9.110873 

io34 

II96 

!358 

i5i9 

1681 

34 

26 

94oo 

9587, 

9774 

996o 

.i47 

.333 

33 

26 

1842 

2003 

2i65 

2326 

2487 

2648 

33 

27 

J.o5o5i9 

o7o6 

0892 

1078 

1264 

i45o 

32 

27 

2809 

297o 

3i3i 

3292 

3453 

36i3 

32 

28 

i635 

1821 

2007 

2I92 

2378 

2563 

3i 

28 

3774 

3935 

4o95 

4256 

44i6 

4577 

3i 

29 

2749 

2934 

3119 

33o4 

3489 

3674 

3o 

29 

4737 

4897 

5o57 

52i8 

5378 

5538 

3o 

3o 

3859 

4o44 

4228 

44i3 

4597 

4782 

20 

3o 

5698 

5858 

6oi7 

6177 

6337 

6497 

29 

3i 

4966 

5i5o 

5335  55i9 

57o3 

5887 

28 

3i 

6656 

6816 

6975 

7i35 

7294 

7453 

28 

32 

6o7i 

6254 

6438  6622 

68o5 

6989 

27 

32 

76i3 

7772 

793i 

8o9o 

8249 

84o8 

27 

33 

7I72 

7356 

7539  7722 

79°5 

8088 

26 

33 

8567 

8726 

8884 

9o43 

92O2 

936o 

26 

34 

827I 

8454 

86378820 

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9i85 

25 

34 

95i9 

9677 

9836 

9994 

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25 

35 

9367 

955o 

9732  9914 

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24 

35 

9.  120469 

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24 

36 

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1188 

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23 

36 

1417 

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1732 

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2047 

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23 

37 

i55i 

1732 

1914  2095 

2276 

2457 

22 

37 

2362 

252O 

2677 

2835 

2992 

3i49 

22 

38 

2639 

2820 

3ooi  3i8i 

3362 

3543 

21 

38 

33o6 

3463 

3620 

3777 

3934 

4o9i 

21 

39 

3724 

3904 

4o85  4265 

4445 

4626 

2O 

39 

4248 

44o4 

456i 

4718 

4874 

5o3i 

20 

4o 

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4986 

5i665346 

5526 

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4o 

5i87 

5344 

55oo 

5656 

58i2 

5969 

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4i 

5885 

6o65 

6244  6424 

66o3 

6783 

18 

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6ia5 

6281 

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6593 

6748 

69o4 

18 

42 

6962 

7141 

7820  7499 

7678 

7857 

17 

42 

7060 

7216 

737i 

7527 

7682 

7838 

ll 

43 

80^6 

82i5 

83938572 

875i 

8929 

16 

43 

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83o4 

8459 

86i4 

8770 

16 

44 

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9286 

9464  9642 

9820 

9998 

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44 

8920 

9o8o 

9235 

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9544 

9699 

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45 

9.o7oi76 

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45 

9854 

.  .  .9 

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46 

1242 

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1774 

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2128 

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46 

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47 

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12 

47 

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48 

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3244 

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49 

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49 

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3704 

3858 

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10 

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548o 

5656 

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6007 

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6358 

9 

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4470 

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9 

5i 

6533 

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6883 

7o58 

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7408 

8 

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6i5o 

8 

52 

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7933 

8107 

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7 

52 

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6455 

6607 

676.>|69i2 

7064 

7 

53 

863i 

88o5 

8980 

9i54 

9328 

95o2 

6 

53 

7216 

7368 

7520 

7672 

-7824 

7976 

6 

54 

9676 

985o 

..24 

.i98 

.372 

.545 

5 

54 

8128 

8279 

843  1 

8582 

8734 

8886 

5 

55 

9.080719 

0892 

1066 

I239 

i4i3 

i586 

4 

55 

9o37 

9i88 

934o 

949i 

9642 

9793 

4 

56|"   i759 

1932 

2IO5 

2278 

245i 

2624 

3 

56 

9944 

..96 

.247 

.398 

.548 

•?99 

3 

57 

2797 

2969 

3l42 

33i4 

3487 

3659 

2 

57 

9.i4o85o 

IOOI 

n5i 

1302 

i453 

i6o3 

2 

58 

3832 

4oo4 

4176 

4348 

4520 

4692 

I 

58 

1754 

i9o4 

2o55 

2205 

2355 

25o5 

1 

59 

4864 

5o36 

5208 

538o 

555i 

5723 

0 

59 

2655 

2806 

2956 

3io6 

3256  34o5 

0 

60" 

50" 

40" 

30"  1  20" 

10" 

, 

60"     50"  |  40"   30"   20"   10" 

d 

Co-sine  of  83  Degrees. 

s 

Co-sine  of  82  Degrees. 

ii 

p  p   (  1"  2"  3"  4"  5"  6"  7"  8'  9" 

(  I"  o"  3'  4"  5"  6"  7"  8"  9" 

irlj  18  37  55  74  92  111  129  148  166 

irt}  16  32  48  G4  80  96  112  128  144 

LOGARITHMIC    TANGENTS. 


1 

Tangent  of  6  Degrees. 

a 

Tangent  of  7  Degrees. 

33 

0" 

10" 

20" 

30" 

40" 

50" 

ii 

0" 

10" 

20" 

30" 

40" 

50"  1 

O 

9.021620 

1823 

2025 

2227 

243o 

2632 

59 

o 

9.089144 

9318 

9492 

9666 

9839 

..i3]59 

I 

2834 

3o36 

3238 

3439 

364i 

3843 

58 

i 

9.o9oi87 

o36i 

o534 

0708 

0881 

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2 

4o44 

4245 

4447 

4648 

4849 

5o5o 

57 

2 

1228 

i4oi 

i574 

I747 

1920 

2093 

57 

3 

525i 

5452 

5653 

5853 

6o54 

6254 

56 

3 

2266 

2439 

2612 

2784 

2957 

3129 

56 

4    64-65 

6655 

6855 

7o55 

7255 

7455 

55 

4 

33o2 

3474  3647 

38i9 

399i 

4i63 

55 

.. 

D 

7655 

7855 

8o55 

8254 

8454 

8653 

54 

5 

4336 

45o8 

468o 

485i 

5o23 

5ig5 

54 

6 

8852 

9052 

9251 

945o 

9649 

9848 

53 

6 

5367 

5538 

57io 

588i 

6o53 

6224 

53 

7 

9.o3oo46 

0245 

o444 

o64a 

0841 

io39 

52 

7 

6395 

6567 

6738 

6909 

7o8o 

725l 

52 

8 

i237 

i435 

i633 

i83i 

2029 

2227 

5i 

8 

7422 

7593 

7764 

7934 

8io5 

8276 

5i 

9 

2425 

2623 

2820 

3017 

32i5 

3412 

5o 

9 

8446 

8616 

8787 

8957 

9127 

9298 

5o 

10 

3609 

38o6 

4oo3 

4200 

4397 

4594 

49 

IO 

9468 

9638 

9808 

9978 

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49 

ii 

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4987 

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ii 

9.  100487 

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12 

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6948 

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12 

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2012 

2181 

235o 

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7i44 

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7535 

773o 

7926 

8121 

46 

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2519 

2688 

2837 

3026 

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3363 

46 

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83i6 

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45 

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9485 

9679 

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16 

9.o4o65i 

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1232 

1426 

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16 

555o 

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22OO 

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278o 

42 

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7225 

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18 

2973 

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4i 

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4i3o 

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20 

5284 

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Q 

20 

9559 

9726 

9892 

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39 

21 

6434 

6626 

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7200 

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38 

21 

9.no556 

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1219 

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38 

22 

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37. 

22 

i55i 

1716 

1882 

2047 

22l3 

2378 

37 

23 

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8917 

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36 

23 

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24 

9869 

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35 

24 

3533 

3698 

3863 

4028 

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26 

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34 

25 

452i 

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34 

26 

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2522 

2711 

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26 

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5999 

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27 

3277 

3466 

3654 

3843 

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32 

27 

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3i 

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27 

32 

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1701 

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27 

33 

9.060016 

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26 

33 

2348 

25lO 

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2833 

2994 

3i56 

26 

34 

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25 

34 

33i7 

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364o 

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2979 

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23 

36 

5249 

5409 

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573o 

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23 

37 

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4821 

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22 

37 

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637i 

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38 

5556 

5739 

5922 

6106 

6289 

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21 

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21 

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6838 

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7387 

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4o 

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17 

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9.  071027 

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43 

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2836 

3oi6 

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44 

2893 

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45 

3839 

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46 

4784 

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47 

5356 

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6074 

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12 

47 

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12 

48 

6432 

6611 

6790 

6969 

7i48 

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II 

48 

6667 

6823 

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49 

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49 

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5o 

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5o 

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9 

5i 

9644 

9822 

.... 

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.355 

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8 

5i 

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9788 

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.254 

8 

52 

9.080710 

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1064 

1241 

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7 

52 

9.i4o4o9 

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7 

53 

i773 

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2480 

2657 

6 

53 

i34o 

i495 

i65o 

i8o5 

i959 

2Il4 

6 

54 

2833 

3oio 

3i86 

3362 

3539 

37i5 

5 

54 

2269 

2424 

2578 

2733 

2887 

3o4s 

5 

55 

3891 

4067 

4243 

44i9 

4595 

477i 

4 

55 

3196 

335o 

35o4 

3659 

38i3 

3967 

4 

56 

4947 

5l22 

5298 

5473 

5649'5824 

3 

56 

4l2I 

4a75 

4429 

4583 

4737 

489o 

3 

57 

6000 

6175 

635o 

6525 

6700  6875 

2 

57 

5o44 

5i98 

535! 

55o5 

5659 

58  1  2 

2 

58 

7o5o 

7225 

74oo 

7574 

7749  7924 

I 

58 

5966 

6n9 

6272 

6425 

6579 

6"32 

I 

J2 

8098 

8273 

8447 

8621 

8795  897o 

O 

59 

6885 

7o38 

7191 

7344 

7497  7°5o 

O 

60"    |  50" 

40" 

30" 

20"   10" 

. 

60" 

50" 

40" 

30" 

20"   10" 

ii 

Co-tangent  of  83  Degrees. 

X 

Co-tangent  of  82  Degrees. 

i 

p  p^5  r/  2"  3//  4"  5//  6"  7"  8"  9// 

p  P^t*  l"  2"  3//  4"  5"  6"  7"  8"  9" 

irt£  19  37  56  75  94  110  131  150  1G8 

m\  16  33  49  65  81  98  114  130  146 

LOGARITHMIC    b  i  N  E  s. 


g       Sine  of  8  Degrees. 

.a 

Sine  of  9  Degrees. 

i  a 

0' 

10" 

20" 

30" 

40" 

50* 

s 

0"    1  10"   20" 

30" 

40" 

w 

n 

9,i43i>55 
4453 

3yo5 

46o3 

3855 
4752 

4oo5 
4902 

4i54 
5o5i 

43o4 

5200 

59 

58 

0 

i 

9.  i94332  4465 
5i29  5262 

4598 
5395 

473i 
5527 

4864 
566o 

4997 
5792 

59 

58 

2 

5349 

5498 

5648  5797 

5946  6o95 

57 

2 

5925  60576189 

6322 

6454 

6586 

57 

3 

6243 

6392 

654i  6690 

6839(6987 

56 

Q 

6719 

685i 

6983 

7n5 

7^47 

7379|56 

4 

7i36 

7284 

7433  7581 

773o 

7878 

55 

4 

75n 

7643 

7775 

79°7 

8o38 

8I70J55 

5 

8026 

8174 

832384?! 

86i9 

8767 

54 

c 

83o2 

8434 

8565 

8697 

8828 

896o 

54 

6 

8916 

9063 

9211 

9358 

95o6 

9654 

53 

6 

9091 

9223 

9354 

9486 

96i7 

9748 

53 

7 

9802 

9949  .  .97 

.244 

.392 

.539 

52 

7 

9879 

..II 

.  142 

.273 

•  4o4 

.535 

52 

8 

9.i5o686 

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0981 

1128 

I275 

1422 

5i 

8 

9.200666 

°797 

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io59 

n89 

I  320 

5i 

9 

i569 

1716 

i863 

20IO 

2l57 

23o4 

5o 

9 

i45i 

i582 

1712 

i843 

i973 

2104 

5o 

10 

245i 

2597 

2744 

2891 

3o37 

3i84 

49 

10 

2234 

2365 

2495 

2626 

2756 

2886 

49 

ii 

333o 

3476 

3623 

3769 

39i5 

4o6i 

48 

ii 

3017 

3i47 

3277 

34o7 

3537 

3667 

48 

12 

4208 

4354 

45oo  4646 

4792  4938 

47 

12 

3797 

3927 

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566658i2 

46 

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45 

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45 

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683o 

6975 

7120  7265 

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44 

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44 

16 

7700 

7845 

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43 

16 

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43 

17 

8569 

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18 

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20 

9992 

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39 

21 

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36 

23 

2291 

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34 

25 

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•'092 

7228 

7363 

7498 

7633 

7768 

8 

5i 

3l72 

3293 

34i5 

3536 

3657 

3778 

8 

52 

'.qo3 

8o38 

8i73 

83o8 

8442 

8577 

7 

52 

3899 

4020 

4i4i 

4262 

4383 

45o4 

7 

53 

&,I2 

8847 

8981 

9116 

925o 

9385 

6 

53 

4625 

4746 

4867 

4987 

5io8 

5229 

6 

54 

96.9 

9654 

9788 

9923 

..57 

.191 

5 

54 

5349 

5470 

559i 

57n 

5832 

5952 

5 

55 

9.  I903i5 

o46o 

o594 

0728 

0862 

0996 

4 

55 

6o73 

6i93 

63i3 

6434 

6554 

6674 

4 

56 

n3o 

1264 

i398 

i532 

i665 

1799 

3 

56 

6795 

69i5 

7o35 

7i55 

7275 

7395 

3 

57 

1933 

2066 

2200 

2334 

2467 

2601 

2 

57 

75i5 

7635 

7755 

7875 

7995 

8n5 

2 

58 

2734 

2868 

3ooi 

3i34 

3268 

34oi 

I 

58 

8235 

8355 

84?4 

8594 

87i4 

8834 

I 

59 

3534 

3667 

38oo 

3933 

4o66 

4199 

O 

59 

8953 

9o73 

9I92 

93l2 

943i 

955i 

O 

60" 

50" 

40" 

30"   20" 

10" 

. 

60" 

50" 

40" 

30" 

20" 

10" 

d 

Co-sine  of  81  Degrees. 

3- 

Co-sine  of  80  Degrees. 

§ 

P  l'art$  l"  2"  3"  1"  5"  6"  ""  8"  9" 
H.  lartj  H  28  42  5fi  ?0  g5  9J)  H3  127 

P  P  rt5  l"  2"  3"  4"  5"  6"  7"   8"    ^' 

"•*•"{  13  25  38  50  63  75  88  101  US 

LOGARITHMIC    TANGENTS. 


1 

Tangont  of  S  Degrees. 

1 

* 

Tangent  of  9  Degrees. 

ii 

0" 

10" 

20"   30" 

40" 

50" 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.147803 

7955 

8io8|826i 

84i3 

8566 

59 

O 

9.i997i3 

9849 

9985 

.  121 

.257 

.393 

59 

I 

8718 

8871 

9O23 

9i75 

9328 

948o 

58 

I 

9.2O0529 

o665 

0801 

°937 

io73 

I209 

58 

2 

9632 

9784 

9936 

.88 

.240 

.392 

57 

2 

1  345 

i48i 

1616 

I752 

1888 

2023 

57 

c 

9.i5o544 

o696 

o848 

0999 

n5i 

i3o3 

56 

2 

2l59 

2294 

243o 

2565 

2701 

2836 

56 

4 

i454 

1606 

i757 

I9o9 

2060 

2211 

55 

4 

297I 

3107 

3242 

3377 

35i2 

3647 

55 

r, 

2363 

25i4 

2665 

2816 

2967 

3n8 

54 

c 

3782 

39i8 

4o53 

4i88 

4322 

4457 

54 

6 

3269 

3420 

3571 

3722 

3873 

4023 

53 

6 

4592 

4727 

4862 

4996 

5i3i 

5266 

53 

7 

4174 

4325 

4475 

4626 

4776 

4926 

52 

7 

54oo 

5535 

5669 

58o4 

5938 

6o73 

52 

8 

5077 

5227 

5377 

5528 

5678 

5828 

5i 

8 

6207 

6342 

6476 

6610 

6744 

6878 

5i 

9 

5978 

6128 

6278 

6428 

6578 

6728 

5o 

9 

7oi3 

7147 

7281 

74i5 

7549 

7683 

5o 

IO 

6877 

7027 

7177 

7326 

7476 

7625 

49 

IO 

7817 

795o 

8o84 

8218 

8352 

8485 

49 

ii 

7775 

7924 

8o74 

8223 

8372 

852i 

48 

ii 

8619 

8753 

8886 

9O2O 

9i53 

9287 

48 

12 

8671 

8820 

8969 

9n8 

9267 

94i6 

47 

12 

9420 

9554 

9687 

9820 

9954 

..87 

47 

i3 

9565 

97l3 

9862 

.  .11 

.160 

.3o8 

46 

i3 

9.210220 

o353 

o486 

o6i9 

0752 

o885 

46 

:4 

9.  160457 

o6o5 

o754 

O9O2 

io5i 

1199 

45 

i4 

1018 

u5i 

1284 

1417 

i55o 

i683 

45 

i5 

i347 

i496 

1  644 

I792 

i94o 

2088 

44 

i5 

i8i5 

i948 

2081 

22l3 

2346 

2478 

44 

16 

2236 

2384 

2532 

2680 

2828 

2975 

43 

16 

2611 

2743 

2876 

3oo8 

3i4i 

3273 

43 

17 

3i23 

3271 

34i8 

3566 

37i3 

386i 

42 

iy 

34o5 

3537 

367o 

3802 

3934 

4o66 

42 

18 

4oo8 

4i56 

43o3 

445o 

4598 

4745 

4i 

18 

4198 

433o 

4462 

4594 

4726 

4858 

4i 

i9 

4892 

5o39 

5i86 

5333 

548o 

5627 

4o 

i9 

4989 

5l2I 

5253 

5385 

55i6 

5648 

4o 

20 

5774 

5920 

6067 

6214 

636i 

65o7 

39 

20 

578o 

59n 

<5o43 

6i74 

63o5 

6437 

39 

21 

6654 

6800 

6947 

7o93 

7240 

7386 

38 

21 

6568 

6700 

683i 

6962 

7093 

7225 

38 

22 

7532 

7678 

7825 

7971 

8lI7 

8263 

37 

22 

7356 

7487 

-7618 

7749 

7880 

8011 

37 

23 

84o9 

8555 

8701 

8847 

8992 

9i38 

36 

23 

8142 

8273 

84o3 

8534 

8665 

8796 

36 

24 

9284 

943o 

9575 

972i 

9866 

.  .  12 

35 

24 

8926 

9°57 

9i88 

93i8 

9449 

9579 

35 

25 

9.  170157 

o3o3 

o448 

o593 

o739 

0884 

34 

25 

9710 

984o 

997i 

.101 

.231 

.36i 

34 

26 

1029 

1174 

i3i9 

i464 

i6o9 

i754 

33 

26 

9.220492 

0622 

0752 

0882 

IOI2 

1142 

33 

27 

i899 

2044 

2188 

2333 

2478 

2623 

32 

27 

1272 

1402 

i532 

1662 

1792 

I922 

32 

28 

2767 

29I2 

3o56 

32OI 

3345 

3489 

3i 

28 

2O52 

2l82 

23ll 

2441 

2571 

2700 

3i 

29 

3634 

3778 

3922 

4067 

4211 

4355 

3o 

29 

283o 

2959 

3o89 

32i8 

3348 

3477 

3o 

3o 

4499 

4643 

4787 

493i 

5075 

52i8 

29 

3o 

36o7 

3736 

3865 

3994 

4124 

4253 

29 

3i 

5362 

55o6 

565o 

5793 

5937 

6080 

28 

3i 

4382 

45n 

464o 

4?69 

4898 

5027 

28 

32 

6224 

6367 

65n 

6654 

6797 

694i 

27 

32 

5i56 

5285 

54i4 

5543 

5671 

58oo 

27 

33 

7084 

7227 

737o 

75i3 

7656 

78oo 

26 

33 

5929 

6o58 

6186 

63i5 

6443 

6572 

26 

34 

7942 

8o85 

8228 

837i 

85i4 

8657 

25 

34 

67oo 

6829 

6957 

7086 

7214 

7342 

25 

35 

8799 

8942 

9o85 

9227 

937o 

95l2 

24 

35 

747i 

7599 

7727 

7855 

7983 

8m 

24 

36 

9655 

9797 

9939 

..82 

.224 

.366 

23 

36 

8239 

8368 

8496 

8623 

875i 

8879 

23 

37 

9.i8o5o8 

o65o 

o792 

o934 

1076 

1218 

22 

37 

9°°7 

9i35 

9263 

939o 

95i8 

9646 

22 

38 

i36o 

i5o2 

1  644 

i786 

I927 

2o69 

21 

38 

9773 

99oi 

..29 

.166 

.284 

•  4n 

21 

39 

2211 

2352 

2494 

2635 

2777 

29i8 

20 

39 

9.23o539 

0666 

o793 

092I 

io48 

n75 

2O 

4o 

3o59 

32OI 

3342 

3483 

3625 

3766 

I9 

4o 

1302 

i43o 

i557 

1  684 

1811 

i938 

I9 

4i 

39o7 

4o48 

4i89 

433o 

4471 

4612 

18 

4i 

2o65 

2I92 

2319 

2446 

2573 

2699 

18 

42 

4752 

4893 

5o34 

5i75 

53i5 

5456 

i7 

42 

2826 

2953 

3o8o 

32o6 

3333 

346o 

17 

43 

5597 

5737 

5878 

6018 

6i58 

6299 

16 

43 

3586 

3713 

3839 

3966 

4092 

42I9 

16 

44 

6439 

6579 

6720 

6860 

7000 

7i4o 

i5 

44 

4345 

4471 

4598 

4724 

485o 

4976 

i5 

45 

7280 

7420 

756o 

77oo 

784o 

798o 

1  4 

45 

5io3 

5229 

5355 

548  1 

56o7 

5733 

i4 

46 

8120 

8259 

8399 

8539 

8678 

8818 

i3 

46 

5859 

5985 

6m 

6237 

6362 

6488 

i3 

47 

8958 

9°97 

9236 

9376 

95i5 

9655 

12 

47 

6614 

6740 

6865 

699i 

7117 

7242 

12 

48 

9794 

9933 

..72 

.212 

.35i 

.49o 

II 

48 

7368 

7493 

76i9 

7744 

787o 

7995 

II 

49 

9,  190629 

0768 

o9o7 

io46 

n84 

i323 

IO 

49 

8120 

8246 

837i 

8496 

8621 

8747 

IO 

5o 

1462 

1601 

i739 

i878 

2017 

2i55 

9 

5o 

8872 

8997 

9I22 

9s47 

9372 

9497 

9 

5i 

2294 

2432 

257I 

2709 

2848 

2986 

8 

5i 

9622 

9747 

9872 

9996 

.121 

.246 

8 

52 

3l24 

3262 

3401 

3539 

3677 

38i5 

7 

52 

9.  24o37I 

o495 

O62O 

o745 

o869 

o994 

7 

53 

3953 

4o9i 

4229 

4367 

45o5 

4642 

6 

53 

1118 

1243 

i367 

1492 

1616 

i74i 

6 

54 

4780 

49i8 

5o56 

5i93 

533i 

5468 

5 

54 

i8f5 

i989 

2Il4 

2238 

2362 

2486 

5 

55 

56o6 

5743 

588i 

6018 

6i56 

6293 

4 

55 

26/0 

2734 

2858 

2982 

3io6 

323o 

4 

56 

643o 

6567 

67o5 

6842 

6979 

-71  16 

3 

56 

3354 

3478 

3602 

3726 

385o 

3974 

3 

57 

7253 

739Q 

7527 

7664 

7801 

7938 

2 

57 

4o97 

4221 

4345 

4468 

4592 

i 

58    8o74 

821  1 

8348 

8484 

8621 

8758 

I 

58 

4839 

4962 

5o86 

52O9 

5333 

5456 

i 

59|    8894 

9o3i 

9i67 

93o4 

944o 

9576 

O 

59 

5579 

5703 

5826 

5949 

60-72 

6i96 

o 

60" 

50"   40" 

30"  |  20" 

10" 

g 

GO" 

50" 

40" 

30" 

20" 

ID" 

3* 

Co-tangent  of  81  Degrees. 

Co-tangent  of  80  Degrees. 

p  p  ,  <  I"  2"  3"  4"  5"  6"  7"  8"  9" 

,  p  t  f  1"  2"  3"  4"  5"  6"  7"  8"  9" 

)  H  29  43  58  72  88  101  115  130 

in\  13  26  39  52  65  73  91  103  116 

LOGARITHMIC    Jb  i  IN  E  s. 


1 

Sine  of  10  Degrees. 

d 

Sine  of  1  1  Degrees. 

s 

0" 

10" 

20' 

30" 

40" 

SO' 

8 

0" 

10" 

20"  ]~~30"~P40" 

50" 

o 

9.23967o 

979° 

99°9 

..28 

.i48 

.267 

59 

0 

9.280599 

0707 

o8i5 

0924 

1082 

n4o 

09 

i 

9.24o386 

o5o5 

0624 

o744 

o863 

0982 

58 

I 

1248 

i356 

i465 

i573 

1681 

i789 

58 

2 

IIOI 

1220 

i339 

1  458 

i576 

i695 

57 

2 

1897 

2O05 

2Il3 

2220 

2328 

2436 

5? 

3 

1814 

i933 

2052 

2I70 

2289 

2408 

56 

3 

2544 

2652 

276o 

2867 

2975 

3o83 

56 

4 

2526 

2645 

2763  2882 

3ooi 

3lIQ 

55 

4 

3190 

3298 

34o6 

35i3 

362i 

3728 

55 

5 

3237 

3356 

3474 

3592 

37n 

382Q 

54 

5 

3836 

3943 

4o5i 

4i58 

4266 

4373 

54 

6 

3947 

4o65 

4i84 

4302 

4420 

4538 

53 

6 

448o 

4588 

4695 

4802 

4909 

5017 

53 

7 

4656 

4774 

4892 

5oio 

5i28 

5245 

52 

7 

5i24 

523i 

5338 

5445 

5552 

5659 

52 

8 

5363 

548i 

5599 

57i7 

5834 

5952 

5i 

8 

5766 

5873 

5980 

6o87 

6194 

63oi 

5i 

9 

6069 

6i87 

63o5 

6422 

654o 

6657 

5o 

9 

64o8 

65i4 

6621 

6-728 

6835 

6941 

5o 

10 

6775 

6892 

7oo9 

7I27 

7244 

736i  |49 

IO 

7048 

7i55 

726l 

7368 

7474 

758i 

49 

ii 

7478 

7596 

77i3 

783o 

7947 

8o64 

48 

ii 

7688 

7794 

7900 

8oo7 

8n3 

8220 

48 

12 

8181 

8298 

84i5 

8532 

8649 

8766 

47 

12 

8326 

8432 

8539 

8645 

875i 

8857 

47 

i3 

8883 

8999 

9n6 

9233 

935o 

9466 

46 

i3 

8964 

9070 

9i76 

9282 

9388 

9494 

46 

i4 

9583 

97oo 

98i6 

9933 

..49 

.166 

45 

i4 

9600 

9706 

9812 

9918 

..24 

.i3o 

45 

i5 

9.250282 

0399 

o5i5 

o63i 

o748 

o864 

44 

i5 

9.290236 

o447 

o553 

0659 

o765 

44 

16 

0980 

io97 

I2l3 

1329 

i445 

i56i 

43 

16 

0870 

0976 

1082 

n87 

1293 

i398 

43 

17 

i677 

i793 

I9o9 

2O25 

2l4l 

2257 

42 

17 

i5o4 

1610 

i7i5 

1820 

1926 

2031 

42 

18 

2373 

2489 

26o5  272O 

2836 

2952 

4i 

18 

2137 

2242 

2347 

2453 

2558 

2663 

4i 

19 

3o67 

3i83 

3299  34i4 

353o 

3645 

4o 

i9 

2768 

2874 

2979 

3o84 

3189 

3294 

4o 

20 

376i 

3876 

3992  4io7 

4223 

4338 

39 

20 

3399 

35o4 

3609 

37i4 

38i9 

39s4 

39 

21 

4453 

4568 

4684^799 

49i4 

5029 

38 

21 

4i34 

4239 

4344 

4448 

4553 

38 

22 

5i44 

5259 

53745490 

56o4 

57i9 

37 

22 

4658 

4763 

4867 

4972 

5o77 

6181 

37 

23 

5834 

5949 

6o64  6i79 

6294 

6409 

36 

23 

5286 

539i 

5495 

56oo 

57o4 

58o9 

36 

24 

6523 

6638 

67536867 

6982 

7096 

35 

24 

SgiS 

6017 

6122 

6226 

633o 

6435 

35 

25 

7211 

7326 

744o  7554 

7669 

7783 

34 

25 

6539 

6643 

6747 

6852 

6956 

7060 

34 

26 

7898 

8012 

81268241 

8355 

8469 

33 

26 

7164 

7268 

7372 

7476 

758o 

7684 

33 

27 

8583 

8697 

88118936 

9040 

9i54 

32 

27 

7788 

7892 

7996 

8100 

8204 

83o8 

3a 

28 

9268 

9382 

9495  9609 

9723 

9837 

3i 

28 

8412 

85i5 

8619 

8723 

8827 

893o 

3i 

29 

9951 

..65 

.178 

.292 

.4o6 

.5i9 

3o 

29 

90*54 

9i38 

9241 

9345 

9448 

QQ32 

3o 

3o 

9.26o633 

o747 

0860 

o974 

io87 

I2OI 

29 

3o 

9655 

9759 

9862 

9966 

..69 

.172 

29 

3i 

i3i4 

1428 

i54i  i654 

i768 

1881 

28 

3i 

9.  300276 

o379 

0482 

o586 

0689 

0792 

28 

32 

i994 

2IO7 

2220  2334 

2447 

256o 

27 

32 

o895 

0999 

IIO2 

1205 

i3o8 

i4n 

27 

33 

2673 

2786 

2899  3oi2 

3i25 

3238 

26 

33 

i5i4 

1617 

I720 

1823 

1926 

2029 

26 

34 

335i 

3464 

35763689 

38o2 

39i5 

25 

34 

2l32 

2235 

2337 

2440 

2543 

2646 

25 

35 

4027 

4i4o 

42534365 

4478 

459o 

24 

35 

2748 

285i 

2954 

3o57 

3i59 

3262 

24 

36 

4703 

48i5 

4928;5o4o 

5i53 

5265 

23 

36 

3364 

3467 

3569 

3672 

3774 

3877 

23 

37 

5377 

5490 

56o2(57i4 

5827 

5939 

22 

37 

3979 

4082 

4i84 

4287 

4389 

4491 

22 

38 
39 

6o5i 
6723 

6i63 
6835 

6275 
6947 

6387 
7°59 

6499 
7171 

6611 
7283 

21 
2O 

38 
39 

4593 
5207 

4696 
5309 

4798 
54n 

4900 

5OO2 

56i5 

5io4 
57i7 

21 
20 

4o 

7395 

75o6 

76i8 

773o 

7953 

19 

4o 

58i9 

5921 

6o23 

6i25 

6227 

6328 

I9 

4i 

8o65 

8i76 

8288 

8399 

85n 

8622 

18 

4i 

643o 

6532 

6634 

6736 

6837 

6939 

18 

42 

8734 

8845 

8957 

9068 

9i79 

929I 

17 

42 

704  1 

7142 

7244 

7346 

7447 

7549 

17 

43 

9402 

95i3 

9624 

9736 

9847 

9958 

16 

43 

765o 

7752 

7853 

7955 

8o56 

8i58 

16 

44 

9.270o69 

0180 

O29I 

04O2 

o5i3 

0624 

i5 

44 

8259 

836o 

8462 

8563 

8664 

8766 

i5 

45 

o735 

o846 

o957 

io67 

1178 

I289 

i4 

45 

8867 

8968 

9069 

9i7o 

92-72 

9373 

i4 

46 

i4oo 

i5io 

1621 

I732 

1842 

i953 

i3 

46 

9474 

9575 

9676 

9777 

9878 

9979 

i3 

47 

2064 

2I74 

2285 

2395 

25o5 

2616' 

12 

47 

9.  3  i  0080 

0181 

0282 

o382 

o483 

o584 

12 

48 

2726 

2837 

2947 

3o57 

3i68 

3278 

II 

48 

o685 

o786 

0886 

098-7 

1088 

1189 

II 

49 

3388 

3498 

36o8 

37i8 

3829 

3939 

IO 

49 

I289 

i39o 

i49o 

i59i 

1692 

1792 

IO 

5o 

4049 

4i59 

4269 

4379 

4489 

4598 

9 

5o 

i893 

I993 

2094 

2I94 

2294 

2395 

9 

5i 

4708 

4818 

4928 

5o38 

5i48 

5257 

8 

5i 

2495 

25g5 

2696 

2796  2896 

2997 

8 

52 

53 

5367 
6o25 

5477 
6i34 

5586  5696 
6243  6353 

58o5 
6462 

6572 

7 
6 

52 

53 

3o97 
3698 

3i97 
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3898 

33973497 

3998  4098 

3597 

6 

54 

6681 

679o 

69oo 

7oo9 

7118 

7227 

5 

54 

4297 

4397 

4497 

459714697 

4797 

5 

55 

7337 

7446 

7555 

7664 

7773 

7882 

4 

55 

4897 

4996 

5096 

5196 

5295 

5395 

4 

56 

7991 

8100 

8209 

83i8 

8427 

8536 

3 

56 

5495 

5594 

5694 

5793 

5893 

5993 

3 

57 

8645 

8753 

8862 

8971 

9o8o 

9188 

2 

57 

6o92 

6192 

6291 

63go 

649o 

6589 

2 

58 

9297 

94o6 

95i4 

973i 

984o 

I 

58 

6689 

6788 

6887 

6986  7o86 

7i85 

I 

59 

9948 

..57 

Ties 

.274 

.382 

•  49i 

O 

59 

7284 

7383 

7482 

7582768i 

7780 

O 

60" 

50"  1  40" 

30" 

20" 

10" 

. 

60"     50-''  |  40"   30"   20"   10" 

j 

Co-sine  of  79  Degrees. 

§' 

Co-sine  of  78  Degrees 

% 

(  1"  2"  3"  4"  5"  6"  7"  8''  9" 

C  1"  2"  3"  4"  5"  6"  7"  8"  9" 

IM'artJ  n  23  34  45  57  cg  go  91  1Q2 

<  10  21  31  41  52  62  72  83  93 

LOGARITHMIC    TANGENTS. 


J 

1 

Tangent  of  10  Degrees. 

d 

Tangent  of  1  1  Degrees. 

§ 

0" 

10"  |  20" 

30" 

40" 

50" 

ii 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.246319 

6442 

6565 

6688 

6811 

6934 

59 

o 

9.288652 

8765 

8877 

8989 

9102 

92l4 

59 

I 

7o5y 

7180 

73o3 

7426 

7548 

7671 

58 

I 

9326 

9438 

9661 

9663 

9775 

9887 

58 

2 

7794 

7917 

8o39 

8162 

8285 

84o7 

57 

a 

9999 

.in 

.223 

.335 

•  447 

.559 

57 

3 

853o 

8652 

8775 

8897 

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9142 

56 

3 

9.290671 

o783 

o895 

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1  1  19 

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56 

4 

9264 

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963i 

9753 

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55 

4 

1  342 

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1789 

1901 

55 

5 

9998 

.  120 

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.364 

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54 

5 

20l3 

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2347 

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2570 

54 

6 

9.250730 

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1218 

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53 

6 

2682 

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3127 

3239 

53 

7 

i46i 

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1826 

1948 

2070 

52 

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8 

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29 

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29 

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3i 

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32 

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3-2 

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9969 

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27 

33 

9.270077 

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20 

33 

9.310399 

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0828 

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26 

34 

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25 

34 

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35 

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35 

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1899 

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36 

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23 

36 

2327 

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37 

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3  1  09 

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22 

37 

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38 

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5629 

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5947 

6o53 

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42 

635i 

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6582 

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68i3 

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6159 

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6371 

6477 

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16 

43 

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44 

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7641 

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45 

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8539 

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8884 

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45 

8o64 

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8275 

838i 

8486 

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46 

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47 

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12 

47 

933o 

9435 

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9645 

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12 

48 

9.280488 

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0717 

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o945 

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1  1 

48 

9961 

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.487 

II 

49 

1174 

1288 

1402 

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1744 

10 

49 

9.320592 

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1012 

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5o 

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5o 

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1746 

9 

5i 

2542 

2656 

2770 

2884 

2998 

3m 

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1861 

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2269 

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8 

52 

3225 

3339 

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3566 

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7 

52 

«479 

2583 

2688 

2793 

2897 

3002 

7 

53 

39o7 

4021 

4i34 

4248 

436i 

4474 

6 

53 

3io6 

3211 

33i5 

3420 

3524 

3628 

6 

54 

4588 

47oi 

48i5 

4928 

5o4i 

5  1  54 

5 

54 

3733 

3837 

394i 

4o46 

4i5o 

4254 

5 

55 

5268 

538i 

5494 

5607 

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5833 

4 

55 

4358 

4463 

4567 

467i 

4775 

4879 

4 

56 

5947 

6060 

6i73 

6286 

6399 

65i2 

3 

56 

4g83 

5o87 

5x9x 

5295 

5399 

55o3 

3 

57 

6624 

6737 

685o 

6963 

7o76 

7189 

n 

57 

5607 

57n 

58i5 

59i9 

6o23 

6127 

2 

58 

73oi 

74i4 

7527 

7639 

7752 

7865 

i 

58 

623i 

6334 

6438 

6542 

6646 

6?49 

I 

59 

7977 

8090 

8202 

83i5 

8427 

854o 

0 

59 

6853 

6957 

7060 

7164 

7267 

737, 

O 

60" 

50"   40" 

30"    20" 

10" 

. 

60"     50" 

40" 

30" 

20" 

10" 

. 

Co-tangent  of  79  Degrees. 

S 

Co-tangent  of  78  Degrees. 

i 

P  p»  1  5  1/;  2"  3"  4"  5"  6"  7"  8"  9" 

,  C  1"  2"  3"  4"  5"  6"  7"  8"  9" 

1  I  12  23  35  47  59  70  82  94  106 

irt}  11  22  32  43  54  65  75  86  97 

LOGARITHMIC    SINES. 


& 

Sine  of  12  Degrees. 

d      Sine  of  13  Decrees. 

8 

0' 

10" 

20"   30*   40" 

50" 

S  |    0" 

10" 

20"  |  30" 

40" 

50" 

o 

2 

9.317879 
84?3 
9066 

7978 
8672 
9i65 

8077,817618275 

8671  8769  8868 
oaGS^SOa  9461 

8374 
8967 
9559 

59 
58 
57 

o 

2 

9.352o88 
2635 
3i8i 

2179 
2726 
3272 

2270 
2817 

3363 

2362 

2908 

3454 

2453 
2999 
3545 

2544 
3090 
3636 

59 
58 
5? 

3 
4 

9658 
9.320249 

9757 
o348 

9855 
0446 

9954 
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0643 

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0742 

56 
55 

3 
4 

3726 
4271 

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55 

5 

o84o 

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54 

5 

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6 

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53 

6 

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5720 

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53 

7 

2019 

2117 

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7 

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8 

2607 

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8 

6443 

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9 

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5o 

9 

6984 

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5o 

10 

9.323780 

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10 

9.357524 

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12 

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9.329599 

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4912 

5oo5 

SogS 

5191 

5283 

5376 

12 

47 

7o35 

7120 

72O6 

7292 

7378 

7464 

12 

48 

5469 

556i 

5654 

5747 

5839 

5g32 

II 

48 

7549 

7635 

772I 

7806 

7892 

7978 

II 

49 

6024 

6117 

6210 

63o2 

6395 

6487 

IO 

49 

8o63 

8i49 

8235 

8320 

84o6 

8491 

10 

5o 

9.346579 

6672 

6764 

6857 

6949 

7041 

9 

5o 

9-378577 

8662 

8748 

8833 

8919 

9004 

9 

5i 

7i34 

7226 

73i8 

74io 

75o3 

7595 

8 

5i 

9o89 

9i75 

9260 

9346 

943  1 

95i6 

8 

52 

7687 

7779 

7871 

7963 

8o56 

8i48 

7 

52 

96oi 

0687 

9772 

9857 

9942 

..28 

7 

53 

8240 

8332 

8424 

85i6 

8608 

8700 

6 

53 

9.38on3 

0198 

0283 

o368 

o454 

o539 

6 

54 

8792 

8884 

8976 

9067 

9l59 

925i 

5 

54 

0624 

0709 

o794 

o879 

0964 

io49 

5 

55 

9343 

9435 

9526 

9618 

9710 

9802 

4 

55 

n34 

1219 

i3o4 

i389 

i474 

i559 

4 

56 

9893 

9985 

••77 

.168 

.260 

!s52 

3 

56 

1  643 

1728 

i8i3 

1898 

1983 

2068 

3 

57 

9.35o443 

o535 

0626 

0718 

0809 

0901 

2 

57 

2162 

2237 

2322 

2406 

2491 

2676 

2 

58 

0992 

1084 

1175 

1266 

i358 

1449 

I 

58 

2661 

2745 

283o 

2914 

2999 

3o84 

I 

59 

i54o 

i632 

1723 

1814 

1906 

1997 

O 

59 

3i68 

3253 

3337 

3422 

35o6 

359i 

0 

60" 

50" 

40" 

30" 

20" 

10" 

. 

60" 

50" 

40" 

30" 

20"   10" 

c 

Co-sine  cf  77  Degrees. 

§ 

Co-sine  of  76  Degrees. 

a 

p  p   5  1"  2"  3"  4"  5"  6"  7"  8"  9" 

5  1"  2"  3"  4"  5"  6"  7"  8"  9" 

111  >  P  10  28  38  47  57  66  76  85 

irt)  9   18  26  35  44  53  61  70  79 

I   H  M  i  U 


TANSENTS. 


1 

Tangent  of  12  Degrees 

.3 

Tangent  of  13  Degrees. 

s 

~~0"   FTo" 

20" 

30" 

40" 

50" 

S 

0" 

10" 

20" 

30" 

40" 

50" 

o 

9.32747517578 

7682 

7785 

7888 

7992 

59 

o 

0.363364 

346o 

3556 

3652 

3748 

3844 

59 

i 

8o95 

8199 

8302 

84o5 

85o9 

8612 

58' 

i 

394o 

4o36 

4i32 

4228 

4324 

4420 

58 

2 

8715 

8819 

8922 

9O25 

9I28 

923l 

57 

2 

45i5 

46n 

4707 

48o3 

4899 

4994 

57 

3 

9334 

9438 

954i 

9644 

9747 

985o 

56 

3 

5o9o 

5i86 

5282 

5377 

5473 

5568 

56 

4 

9953 

..56 

.i59 

.262 

.365 

.468 

55 

4 

5664 

5760 

5855 

595i 

6o46 

6142 

55 

5 

9.33o57o 

0673 

0776 

o879 

0982 

1084 

54 

5 

6237 

6333 

6428 

6524 

6619 

67i5 

54 

6 

1187 

I29O 

i393 

i495 

i598 

1701 

53 

6 

6810 

69o5 

7001 

7o96 

7191 

7287 

53 

7 

i8o3 

I9o6 

2008 

2III 

22l3 

23i6 

52 

7 

7382 

7477 

7572 

7668 

7763 

7858 

52 

8 

2418 

2521 

2623 

2726 

2828 

293o 

5i 

8 

7953 

8o48 

8143 

8239 

8334 

8429 

5i 

9 

3o33 

3i35 

3237 

334o 

3442 

3544 

5o 

9 

8524 

8619 

8714 

88o9 

8904 

8999 

5o 

10 

9.333646 

3  748 

385i 

3953 

4o55 

4i57 

49 

IO 

9.369o94 

9i89 

9284 

9378 

9473 

9568 

49 

ii 

4259 

436i 

4463 

4565 

4667 

4769 

48 

ii 

9663 

9758 

9853 

9947 

..42 

.i37 

48 

12 

4871 

4973 

5o75 

5i77 

5279 

538o 

47 

12 

9.370232 

o326 

0421 

o5i6 

0610 

0705 

47 

i3 

5482 

5584 

5686 

5788 

5889 

599i 

46 

i3 

°799 

o894 

o989 

io83 

1178 

1272 

46 

i4 

6093 

6i94 

6296 

6398 

6499 

6601 

45 

i.4 

i367 

i46i 

i556 

i65o 

1744 

i839 

45 

i5 

6702 

6804 

69o5 

7007 

7108 

7210 

44 

r5 

I933 

2028 

2122 

2216 

23ll 

24o5 

44 

16 

73n 

74i3 

?5i4 

76!5 

7717 

7818 

43 

16 

2499 

2593 

2688 

2782 

2876 

297o 

43 

17 

7919 

8021 

8122 

8223 

8324 

8426 

42 

17 

3o64 

3i59 

3253 

3347 

344i 

3535 

4a 

18 

8527 

8628 

8729 

883o 

893i 

9o32 

4i 

18 

3629 

3723 

38i7 

39n 

4oo5 

4o99 

4i 

'9 

9i33 

9234 

9335 

9436 

9537 

9638 

4o 

19 

4i93 

4287 

438i 

4475 

4569 

4662 

4o 

20 

9.339739 

984o 

994i 

..42 

.i43 

.243 

39 

20 

9.374756 

485o 

4944 

5o38 

5i3i 

5225 

39 

21 

9.34o344 

o445 

o546 

o646 

0747 

o848 

38 

21 

53i9 

54i3 

55o6 

56oo 

5694 

5787 

38 

22 

o948 

io49 

n5o 

1250 

i35i 

i45i 

37 

22 

588i 

5975 

6068 

6162 

6255 

6349 

37 

23 

i552 

i652 

i753 

i853 

i954 

2o54 

36 

23 

6442 

6536 

6629 

6723 

6816 

69io 

36 

24 

2i55 

2255 

2355 

2456 

2556 

2656 

35 

24 

7oo3 

7096 

7i9o 

7283 

7376 

7470 

35 

25 

2757 

2857 

2957 

3o57 

3i58 

3258 

34 

25 

7563 

7656 

775o 

7843 

7936 

8o29 

34 

26 

3358 

3458 

3558 

3658 

3758 

3858 

33 

26 

8122 

8216 

83o9 

8402 

8495 

8588 

33 

27 

3958 

4o58 

4i58 

4258 

4358 

4458 

32 

27 

8681 

8774 

8867 

896o 

9o53 

9i46 

32 

28 

4558 

4658 

4?58 

4858 

4957 

5o57 

3i 

28 

9239 

9332 

9425 

95i8 

96n 

97°4 

3i 

29 

5i57 

5257 

5357 

5456 

5556 

5656 

3o 

29 

9797 

989o 

9983 

..75 

.168 

.261 

3o 

3o 

9.345755 

5855 

5954 

6o54 

6i54 

6253 

29 

3o 

9.38o354 

o446 

o539 

o632 

0725 

0817 

29 

3i 

6353 

6452 

6552 

665i 

675i 

685o 

28 

3i 

o9io 

ioo3 

io95 

1188 

1280 

i373 

28 

32 

6949 

7o49 

7i48 

7248 

7347 

7446 

27 

32 

i466 

i558 

i65i 

1743 

i836 

1928 

27 

33 

7545 

7645 

7744 

7843 

7942 

8042 

26 

33 

2O2O 

2Il3 

2205 

2298 

239o 

2482 

26 

34 

8i4i 

8240 

8339 

8438 

8537 

8636 

25 

34 

2575 

2667 

2759 

2852 

2944 

3o36 

25 

35 

8735 

8834 

8933 

9o32 

9i3i 

9230 

24 

35 

3l29 

3221 

33  1  3 

34o5 

3497 

3589 

24 

36 

9329 

0428 

9527 

9626 

9724 

9823 

23 

36 

3682 

3774 

3866 

3958 

4o5o 

4i42 

23 

3? 

9922 

.  .21 

.  I2O 

.218 

.3i7 

.4i6 

22 

37 

4234 

4326 

44i8 

45io 

4602 

4694 

22 

38 

9.35o5i4 

o6i3 

0712 

0810 

o9o9 

1007 

21 

38 

4786 

4878 

497o 

5o62 

5i53 

5245 

21 

39 

1  1  06 

I2O4 

i3o3 

i4oi 

i5oo 

i5g8 

20 

39 

5337 

5429 

552i 

56i2 

5704 

5796 

2O 

4o 

9.351697 

i795 

i894 

I992 

2O9O 

2189 

19 

4o 

9  385888 

5979 

6071 

6i63 

6254 

6346 

I9 

4i 

2287 

2385 

2483 

2582 

2680 

2778 

iS 

4i 

6438 

6529 

6621 

67I2 

68o4 

6895 

18 

42 

2876 

2974 

3o73 

3i7i 

3269 

3367 

17 

42 

6987 

7078 

7170 

726l 

7353 

7444 

17 

43 

3465 

3563 

366i 

3759 

3857 

SgSS 

16 

43 

7536 

7627 

7718 

78io 

79oi 

7992 

16 

44 

4o53 

4i5i 

4249 

4347 

4445 

4542 

i5 

44 

8084 

8i75 

8266 

8358 

8449 

854o 

i5 

45 

464o 

4738 

4836 

4934 

5o3i 

5  1  29 

i4 

45 

863i 

8722 

88i4 

89o5 

8996 

9o87 

i4 

46 

5227 

5324 

5422 

552O 

56i7 

57i5 

i3 

46 

9i78 

9269 

9360 

945i 

9542 

9633 

r3 

47 

58i3 

59io 

6008 

6io5 

6203 

63oo 

12 

47 

9724 

98i5 

9906 

9997 

..88 

.179 

12 

48 

63o8 

6495 

6593 

669o 

6787 

6885 

II 

48 

9.390270 

o36i 

0452 

o543 

o633 

0724 

II 

49 

6982 

7°79 

7177 

7274 

737i 

7469 

IO 

49 

o8i5 

o9o6 

0997 

io87 

1178 

1269 

IO 

5o 

9.357566 

7663 

7760 

7857 

7954 

8o52 

9 

5o 

9.391360 

i45o 

i54i 

i632 

1722 

i8i3 

9 

5i 

8149 

8246 

8343 

844o 

8537 

8634 

8 

5i 

1903 

i994 

2o85 

2I75 

2266 

2356 

8 

5a 

8731 

8828 

8925 

9O22 

9119 

92l6 

7 

52 

2447 

2537 

2628 

27l8 

2808 

2899 

7 

53 

93i3 

94o9 

95o6 

96o3 

9700 

9797 

6 

53 

2989 

3o8o 

3170 

3260 

335i 

344  1 

6 

54 

9893 

999° 

..87 

.184 

.280 

.377 

5 

54 

353i 

3622 

3712 

38o2 

3892 

3983 

5 

55 

9.36o474 

o57o 

0667 

o763 

0860 

o957 

4 

55 

4073 

4i63 

4253 

4343 

4433 

4523 

4 

56 

io53 

n5o 

1246 

1  343 

i439 

i535 

3 

56 

46i4 

4704 

4794 

4884 

4974 

5o64 

3 

57 

i632 

1728 

1825 

I92I 

2017 

2Il4 

2 

57 

5i54 

5244 

5334 

5424 

55i4 

56o4 

2 

58 

22IO 

23o6 

24o3 

2499 

2595 

269I 

I 

58 

5694 

5783 

5873 

5963 

6o53 

6i43 

I 

59 

2787 

2884 

298o 

3076 

3172 

3268 

O 

59 

6233 

6322 

6412 

65o2 

6592 

6681 

O 

60" 

50" 

40" 

30" 

20" 

10" 

fi 

GO" 

50"  1  40" 

30" 

20" 

10" 

fl 

Co-tangent  of  77  Degrees. 

a 

Co-tangent  of  76  Degrees. 

i 

P  p   C  L"  2"  3"  4"  5"  6"  7"  8"  9" 

,  (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

\  10  20  30  40  50  60  70  80  90 

r.rartj  g  lg  „%  37  ^  5fi  65  74  g3 

<•      * 

38 


LOGARITHMIC   SINES*. 


.3 

Sine  of  14  Degrees. 

d 

Sine  of  1  5  Degrees. 

m 

0" 

10" 

20"   30"   40"  |  50" 

9 

0" 

10" 

20" 

30" 

40" 

50" 

1 

0 

9.383675 

3760 

38443928 

4oi3 

4o97 

b9 

o 

0.412996 

3o75 

3i53 

3232 

33io 

3389 

59 

I 

4182 

4266 

435o4435 

45i9 

46o3 

58 

I 

3467 

3546 

3624 

37o3 

378i 

386o 

58 

2 

4687 

4772 

4856:494o 

5  024 

5io8 

57 

2 

39,38 

4oi6 

4o95 

4i73 

4252 

433o 

5? 

c 

5192 

5277 

536i  5445 

5529 

56i3 

56 

2 

44o8 

4486 

4565 

4643 

472I 

48oo 

56 

4 

5697 

578i 

5865  5949 

6o33 

6117 

55 

4 

4878 

4956 

5o34 

5lI2 

5190 

5269 

55 

c 

6201 

6a85 

6369  6452 

6536 

6620 

54 

c 

5347 

5425 

55o3 

558i 

5659 

5737 

54 

6 

6704 

6788 

6872  6955 

7o39 

7123 

53 

6 

58i5 

5893 

597i 

6o49 

6l27 

62o5 

53 

7 

7207 

7290 

7374  7458 

754i 

7625 

52 

7 

6283 

636i 

6439 

65i7 

6595 

6673 

52 

8 

7709 

7792 

78767959 

8o43 

8127 

5i 

8 

675i 

6828 

6906 

6984 

7062 

7i4o 

5i 

9 

8210 

8294 

88778461 

8544 

8627 

5o 

9 

72i7 

7295 

7373 

745i 

7528 

76o6 

5o 

10 

9.388711 

8794 

8878896i 

9044 

9,128 

49 

10 

9.417684 

776i 

7839 

79i7 

7994 

8o72 

49 

ii 

9211 

9294 

9378 

946l 

9544 

9627 

48 

ii 

8i5o 

8227 

83o5 

8382 

846o 

8537 

48 

12 

9711 

9794 

9877 

9960 

..43 

.126 

47 

12 

86i5 

8692 

877o 

8847 

8925 

9002 

47 

i3 

9.390210 

0293 

o376 

0459 

0542 

o625 

46 

i3 

9°  79 

9l57 

9234 

9312 

9389 

9466 

46 

i4 

0708 

0791 

o874 

°957 

1040 

1123 

45 

i4 

9544 

96-21 

9698 

9776 

9853 

993o 

45 

i3 

1206 

1289 

i37i 

i454 

i537 

1620 

44 

i5 

9.420007 

oo85 

0162 

0^39 

o3i6 

o393 

44 

16 

1703 

1786 

1868 

1951 

2o34 

2117 

43 

16 

0470 

o548 

0625 

O702 

°779 

o856 

43 

I7 

2199 

2282 

2365 

2447 

253o 

26i3 

42 

17 

0933 

IOIO 

io87 

n64 

1241 

i3i8 

42 

18 

2695 

2778 

2860 

2943 

3o25 

3io8 

4i 

18 

i395 

l472 

1  549 

1626 

1703 

i78o 

4i 

J9 

3191 

3273 

3356 

3438 

3520 

36o3 

4o 

J9 

1857 

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2OIO 

2087 

2164 

224l 

4o 

20 

9.393685 

3768 

385o 

3932 

4oi5 

4o97 

39 

20 

9.4223i8 

2394 

247I 

2548 

2625 

27OI 

39 

21 

4179 

4262 

4344 

4426 

45o8 

459i 

38 

21 

2778 

2855 

293i 

3oo8 

3o85 

3i6i 

38 

22 

4673 

4755 

4837 

4919 

5oo2 

5o84 

3? 

22 

3238 

33i5 

339i 

3468 

3544 

3621 

37 

23 

5i66 

5248 

533o 

5412 

5494 

5576 

36 

23 

3697 

3774 

385o 

3927 

4oo3 

4o8o 

36 

24 

5658 

5740 

5822 

5904 

5986 

6068 

35 

24 

4i56 

4233 

4309 

4386 

4462 

4538 

35 

25 

6i5o 

6232 

63i4 

6395 

6477 

6559 

34 

25 

46i5 

4691 

4767 

4844 

4920 

4996 

34 

26 

664i 

6723 

68o5 

6886 

6968 

7o5o 

33 

26 

5o73 

5i49 

5225 

53oi 

5378 

5454 

33 

27 

7132 

7213  7295 

7377 

7458 

754o 

32 

27 

553o 

5  606 

5682 

5758 

5835 

59n 

32 

28 

7621 

77o3  7785 

7866 

7948 

8029 

3i 

28 

5987 

6o63 

6139 

621  5 

6291 

6367 

3i 

29 

8111 

8i92'8274 

8355 

8437 

85i8 

3o 

29 

6443 

65i9 

6595 

6671 

6747 

6823 

3o 

3o 

9.398600 

8681 

8762 

8844 

8925 

9oo7 

29 

3o 

9.426899 

6975 

7o5i 

7127 

7202 

7278 

29 

3i 

9088 

9169 

925o 

9332 

94i3 

9494 

28 

3i 

7354 

743o 

75o6 

7582 

7657 

7733 

28 

32 

9575 

9657 

9738 

9819 

9900 

9981 

27 

32 

•7809 

7885 

-7960 

8o36 

8112 

8187 

27 

33 

9.400062 

oi44 

0225 

o3o6 

o387 

o468 

26 

33 

8263 

8339 

84i4 

8490 

8566 

864i 

26 

34 

0549 

o63o 

0711 

o792 

o873 

o954 

25 

34 

87i7 

8792 

8868 

8944 

9019 

9o95 

25 

35 

io35 

1116 

1197 

1277 

i358 

i439 

24 

35 

9i7o 

9246 

9321 

9397 

947a 

9547 

24 

36 

l52O 

1601 

1682 

1762 

i843 

I924 

23 

36 

9623 

9698 

9774 

9849 

9924 

23 

:  37 

2OO5 

2o85 

2166 

2347 

2328 

2408 

22 

37 

9.43oo75 

oi5o 

0226 

o3oi 

o376 

o45i 

22 

-  38 

2489 

257o 

265o 

273l 

2811 

2892 

21 

38 

o527 

0602 

o677 

0752 

0828 

o9o3 

21 

39 

297fi 

3o53 

3i33 

3214 

3294 

3375 

2O 

39 

o978 

io53 

1128 

1203 

I278 

1  354 

20 

4o 

9.4o3455 

3536 

36i6 

3697 

3777 

3857 

J9 

4o 

9.43i429 

i5o4 

i579 

1  654 

I729 

1804 

I9 

4i 

3938 

4oi8 

4098 

4179 

4259 

4339 

18 

4i 

i879 

1954 

2029 

2104 

21-79 

2254 

18 

4a 

4420 

45oo 

458o 

466o 

474i 

4821 

J7 

42 

2329 

24o3 

2478 

2553 

2628 

27o3 

J7 

43 

4901 

498i 

5o6i 

5i4i 

5221 

53o2 

16 

43 

2778 

2853 

292-7 

3OO2 

3o77 

3i52 

16 

44 

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3226 

33oi 

3376 

345  1 

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43 

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5942 

6022 

6102 

6181 

6.261 

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45 

3675 

3749 

3824 

3898 

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46 

634i 

6421 

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6661 

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46 

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7378 

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7538 

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48 

5oi6 

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:  49 

7777 

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49 

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5834 

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;  50 

9.408254 

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52 

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53 

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9920 

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53 

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7464 

7538 

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6 

54 

9.410157 

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5 

54 

7686 

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7834 

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5 

55 

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4 

55 

8i29 

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4 

56 

1106 

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1264 

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1422 

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3 

56 

8572 

8646 

8-719 

8793 

8867 

894i 

3 

*7 

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1816 

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2 

57 

9oi4 

9088 

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9235 

93o9 

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2 

58 

2052 

2l3l 

2210 

2288 

2367 

2446 

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58 

9456 

953o 

96o3 

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9824 

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59 

2524 

26o3 

2682 

276o 

2839 

2918 

0 

59 

9897 

997i 

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.265 

0 

60" 

50" 

40"  I  30" 

20" 

10" 

a 

60" 

50" 

40" 

30" 

20* 

10" 

g 

Co-sine  of  75  Degrees. 

.5 

s 

Co-sine  of  74  Degrees. 

S 

p  P  t  $  1"  2"  3"  4"  5"  6"  7"  8"  9" 

C  1"  2"  3"  4"  5"  6"  7"  8"  9" 

1   **{  8  16  24  33  41  49  57  65  73 

r.rart^  s  ig  23  3Q  3g  46  53  gl  6g 

LOG  \uiTH MIC    TANGENTS. 


7- 

Tangent  of  14  Degrees. 

II  -a 

Tangent  of  15  Degrees. 

0"    |  10"  |  20"   30" 

40"  I  50" 

g 

0" 

10''   20" 

30" 

40"   50" 

0 

I 

2 

3 

9  .39077i  |686i  695o 
730978997488 

7846  793618026 

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7578 
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7i3o 
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59 
58 
57 
56 

0 

i 

2 

3 

9.  428052 
8558 
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4 

9.43oo7o- 

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5 

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9544 

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6 

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2162 

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1680 

1769 

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258o 

2663 

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21 

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29 

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32 

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27 

33 

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33 

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34 

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34 

4947 

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25 

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39 

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20 

39 

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9.417842 

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42 

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43 

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48 

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2064 

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49 

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2633 

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2888 

10 

49 

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5o 

9.433974 

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5o 

9.  4527O6 

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8 

52 

3993 

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4248 

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7 

52 

3668 

3748 

3828 

3908 

3988 

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7 

53 

45o3 

4587 

4672 

4757 

4842 

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6 

53 

4i48 

4228 

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4388 

4468 

4548 

6 

54 

5on 

5o96 

5i8i 

5265 

535o 

5435 

5 

54 

4628 

4708 

4787 

4867 

4947 

5027 

5 

55 

55i9 

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5689 

5773 

5858 

5942 

4 

55 

5107 

5i87 

5267 

5346 

5426 

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4 

56 

6027 

6112 

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6281 

6365 

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3 

56 

5586 

5655 

5745 

5825 

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5984 

3 

57 

6534 

6619 

67o3 

6788 

6872 

6956 

2 

57 

6064 

6i44 

6223 

63o3 

6383 

6462 

2 

58 

7041 

7125 

7210 

7294 

7378 

7463 

I 

58 

6542 

6622 

6701 

6781 

6860 

694o 

I 

59 

7547 

763i 

77i5 

78oo 

7884 

7968 

O 

59 

7019 

7099 

7178 

7258 

733774i7 

O 

60"    |  50"   40" 

30"  |  20"   10" 

d 

60"     50" 

40"   30" 

20"   10" 

a 

Co-tangent  of  75  Degrees. 

•9 

Co-tangent  of  74  Degrees. 

•W 

& 

p  P-,  *J  '"  ~"  3"  4//  5"  (5//  7//  8"  9" 

p  p   <  1"  2"  3"  4<  5"  6"  7"  8"  9;' 

19  17  26  35  43  52  61  69  78 

\  8  lfG  25  33  41  49  57  65  74 

L L 


4  ii 


LOGARITHMIC    SINES. 


c 

Sine  of  16  Degrees. 

d 

Sine  of  1  7  Degrees. 

•^ 

0" 

10" 

20"   30" 

40"   50" 

§ 

0" 

10" 

20" 

30" 

40"   50" 

O 

9-44o338 

o4n 

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59 

O 

9.465935 

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6142 

621  1 

6280 

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0778 

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n45 

58 

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6348 

6417 

6486 

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6623 

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58 

2 

1218 

1292 

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1  584 

57 

2 

676i 

683o 

6898 

6967 

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57 

3 

i658 

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1804 

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2O23 

56 

3 

7i73 

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7379 

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56 

4 

2096 

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2462 

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4 

7585 

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55 

5 

2535 

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2827 

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5 

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6 

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7 

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8 

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8 

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9 

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9 

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10 

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49 

10 

9.47oo46 

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59 

9593 

9658 

9723 

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0 

eo" 

50" 

40" 

30"   20'' 

10" 

a 

60"     50" 

40" 

30" 

20" 

10" 

d 

Co-sine  of  73  Degrees. 

3 

Co-sine  of  72  Degrees. 

1 

P  PartJ  !"  2/  3"  4"  5"  6"  7"  8//  9" 

t  1"  2"  3"  4"  5"  0"  7"  8"  9" 

'  ' 

P.  Part  J  ?  j  3  20  27  33  40  47  53  60 

LOGAR-THMIC      TANGENTS. 


41 


fl 

Tangent  of  16  Degrees. 

d 

Tangent  of  17  Degrees. 

UL 

0" 

10" 

20" 

30" 

40" 

50" 

91 

0" 

10" 

20" 

30" 

40" 

50" 

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60" 

50" 

40" 

30" 

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10" 

a 

tO" 

50" 

40" 

30" 

20" 

10" 

. 

Co-tangent  of  73  Degrees. 

iQ 

s 

Co-tangent  of  72  Degrees. 

i 

P  Pnrf5  1/7  2//  3//  4"  5//  6"  7//  8"  9" 

{ll\  8  15  23  31  39  46  54  62  70 

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t  I/'   c,ll   o//   AH   ell   pll   j//   ot/   nil 
P  Part  )l~<)        OO/O 

)  7  15  22  29  37  44  51  59  66 

v    4                 _ 

L-JGARITHMIC   SINES. 


c  j      Sine  of  18  Degrees. 

d 

Sine  of  19  Degrees. 

2     0" 

10" 

20" 

30"   40" 

50" 

s 

0"      10" 

20" 

30" 

40" 

50" 

o  9.489982 

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42 

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42 

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43 

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44 

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49 

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1265 

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6 

53 

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54 

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54 

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55 

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1049 

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4 

55 

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2370 

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4 

56 

1172 

1233 

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1478 

3 

56 

2661 

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2777 

2835 

2893 

2951 

3 

57 

i54o 

1601 

1662 

1724 

1785 

1  846 

2 

57 

3oo9 

3o67 

3i25 

3i83 

324i 

3299 

2 

58 

1907 

1969 

2o3o 

209I 

2l52 

2214 

I 

58 

3357 

34i5 

3473 

353i 

3589 

3647 

I 

59 

2276 

2336 

2397 

2458 

2520 

258i 

O 

59 

3  7o4 

3762 

3820 

3878 

3936 

3994 

0 

60" 

50" 

40" 

30" 

20" 

10" 

C 

60" 

50" 

40" 

30" 

20" 

10" 

j 

Co-sine  of  7  1 

Degrees. 

.5 

Co-sine  of  70  Degrees. 

3 

f  til  nil   q//   AH 
P  Part)             4 

irt{  6  13  19  25 

5-'  6"  7"  8"  9" 
31  38  44  50  57 

,.(!"  2"  3"  4"  5"  6"  7"  8"  9"  1 
irl\  6  12  18  24  30  36  42  48  54  | 

LOGARITHM    j    TANGENTS. 


.5 

Tangent  of  18  Degrees. 

J 

Tangent  of  1  9  Degrees. 

2 

0' 

10" 

20" 

30" 

40" 

50" 

2 

0" 

10" 

20" 

30" 

40" 

50*^ 

o 

9.511776 

1  848 

1919 

1991 

2o63 

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59 

o 

9.536972 

7040 

7109 

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2206 

2277 

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2492 

2564 

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i 

7382 

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2 

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57 

2 

7792 

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3 

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3421 

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3 

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4 

3493 

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55 

4 

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5 

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6 

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52 

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54 

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5 

54 

87o3 

8768 

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8966 

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5 

55 

4916 

4985 

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5259 

4 

55 

9°97 

9163 

9229 

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9360 

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4 

56 

5328 

5396 

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5534 

5602 

567i 

3 

56 

949i 

9557 

9623 

9688 

9754 

9820 

3 

57 

5739 

58o8 

5876 

5945 

6oi3 

6082 

2 

57 

9885 

9951 

..17 

..82 

.i48 

.214 

9 

58 

6i5o 

6219 

6287 

6356 

6424 

6493 

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58 

9.56o279 

o345 

o4io 

o476 

o542 

o6o7 

I 

59 

656i 

663o 

6698 

6767 

6835 

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O 

59 

0673 

0738 

o8o4 

o869 

09  3  5 

1000 

O 

60" 

50" 

40" 

30" 

20" 

10" 

C 

60"     50" 

40" 

30" 

20" 

10" 

. 

Co-tangent  of  71  Degrees. 

ifl 

8 

Co-tangent  of  70  Degrees. 

a 

ppnrt$  1"  2"  3"  4"  5"  6"  7"  8"  9" 

.(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

"{  7  14  21  28  35  42  4.9  56  63 

in\  7  13  20  17  33  40  47  54  60 

LOGARITHMIC    SINES 


1 

Sine  of  20  Degrees. 

.5 

Sine  of  21  Decrees. 

93 
l- 

*Tg 

10" 

20" 

30" 

40" 

50" 

s 

0" 

10"  |  ao" 

30" 

40" 

50" 

o 

9.534o52 

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59 

0 

9.55432Q 

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r 

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58 

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2 

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57 

2 

4987 

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57 

3 

5o92 

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56 

3 

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4 

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4 

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5    5783 

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5 

5971 

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54 

61    6:29 

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53 

6 

6299 

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53 

7 

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52 

7 

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6473 

6526 

6579 

22 

38 

7019 

7o75 

7i3i  7187 

7242 

7298 

21 

38 

6632 

6685 

6739 

6792 

6845 

6898 

21 

39 

7354 

7410 

7466  7522 

7578 

7633 

20 

39 

695i 

7004 

7o57 

7no 

7i63 

72l6 

2O 

4o  9.547689 

7745 

7801)7857 

79I2 

7968 

19 

4o 

9.567269 

7322 

7375 

7428 

748i 

7534 

I9 

4i 

8024 

8080 

81368191 

8247 

83o3 

18 

4i 

7587 

7640 

7693 

7746 

7799 

785i 

18 

42 

8359 

84i4 

847o'8526 

858i 

8637 

i? 

42 

79°4 

7957 

8010 

8o63 

8116 

8i69 

I7 

43 

8693 

8748 

88o4 

8860 

89i5 

897i 

16 

43 

8222 

8275 

8327 

838o 

8433 

8486 

16 

44 

9027 

9082 

9i38 

9I93 

9249 

93o5 

i5 

44 

8539 

8592 

8644 

8697 

875o 

88o3 

i5 

45 

9360 

9416 

9471 

95a7 

9582 

9638 

i4 

45 

8856 

8908 

896i 

9oi4 

9o67 

9ii9 

i4 

46 

9693 

9749 

98o5 

9860  9916 

997i 

i3 

46 

9172 

9225 

9277 

933o 

9383 

9436 

i3 

47  9.55oO26 

0082 

0137 

0193  0248 

o3o4 

12 

47 

9488 

954i 

9594 

9646 

9699 

9752 

12 

48    o359 

o4i5 

0470 

o535 

o58i 

o636 

II 

48 

9804 

9857 

99io 

9962 

..i5 

..67 

II 

49]    0692 

0747 

0802 

o858 

o9i3 

o968 

IO 

49 

9.570120 

0173 

O225 

02-78 

o33o 

o383 

IO 

5o'9.55io24 

1079 

n34 

1190 

1245 

i3oo 

9 

5o 

9.570435 

o488 

o54i 

o593 

o646 

o698 

9 

5i    i356 

i4n 

i486 

l52I 

i577 

i632 

8 

5i 

o75i 

o8o3 

o856 

o9o8 

o96i 

ioi3 

8 

52 

1687 

1742 

1798 

i853 

I9o8 

I963 

7 

52 

1066 

1118 

II7O 

1223 

I275 

i328 

7 

53 

2018 

2074 

2129 

2184 

2239 

2294 

6 

53 

i38o 

i433 

i485 

i537 

i59o 

1642 

6 

54 

2349 

24o5 

2460 

25i5 

257O 

2625 

5 

54 

i695 

1747 

i799 

i852 

I9o4 

i956 

5 

55 

2680 

2735 

2790 

2845 

29OO 

2955 

4 

55 

2009 

2061 

2Il3 

2766 

2218 

2270 

4 

56 

3oio 

3o65 

3l2I 

3i76 

323i 

3286 

3 

56 

2323 

2375 

2427 

2479 

2532 

2584 

3 

57 

334i 

3396 

345  1 

35o6 

356o 

36i5 

2 

57 

2636 

2688 

2741 

2793 

2845 

2897 

2 

58 

367o 

3725 

378o 

3835 

389o 

3945 

I 

58 

2g5o 

3002 

3o54 

3io6 

3i58 

3210 

I 

*9 

4ooo 

4o55 

4no 

4i65  4219 

4274 

O 

59 

3263 

33i5 

3367 

34i9 

347i 

3523 

O 

60" 

50" 

40" 

30"  |  20" 

10" 

fl 

60" 

50" 

40" 

30" 

20" 

10" 

d 

Co-sine  of  69  Degrees. 

Co-sine  of  68  Degrees. 

3 

„  -p   (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

T,  -p   (  I"  2"  3"  4"  5"  6"  7"  8"  9" 

F.FartJ  6  n  17  23  28  34  39  45  51 

c 

l.^artj  5  n  16  21  27  32  3?  43  4g 

r  

LOGARITHMIC    TANGENTS. 


J 

Tangent  of  20  Degrees. 

<d 

Tangent  of  21  Degrees. 

2 

0" 

10" 

20" 

30" 

40" 

50" 

s 

0" 

10"   20" 

30" 

40" 

50" 

0 

9.561066 

n3i 

1197 

1262 

i328 

i393 

59 

0 

9-584177 

424o 

43o3 

4366 

4420 

4492 

59 

I 

1459 

i524 

1  5go 

i655 

1721 

1786 

58 

i 

4555 

46i8 

468i 

4744 

48o6 

4869 

58 

2 

i85i 

1917 

1982 

2048 

2Il3 

2178 

57 

2 

4932 

4995 

5o58 

5l2I 

5i83 

5246 

57 

3 

2244 

2309 

2375 

2440 

25o5 

2571 

56 

3 

5309 

5372 

5435 

5498 

556o 

5623 

56 

4 

2636 

2701 

2767 

2832 

2897 

2963 

55 

4 

5686 

5749 

58n 

5874 

5937 

6000 

55 

5 

3o28 

3o93 

3i58 

3224 

3289 

3354 

54 

5 

6062 

6i25 

6188 

625i 

63i3 

6376 

54 

6 

3419 

3485 

355o 

36i5 

368o 

3746 

53 

6 

6439 

65oi 

6564 

6627 

6689 

6752 

53 

7 

38n 

3876 

394i 

4oo6 

4071 

4i37 

52 

7 

68i5 

6877 

6940 

7oo3 

7o65 

7128 

52 

8 

4202 

4267 

4332 

4397 

4462 

4527 

5i 

8 

7190 

7253 

73i6 

7378 

744i 

75o3 

5i 

9 

4593 

4658 

4723 

4788 

4853 

49i8 

5o 

9 

7566 

7629 

7691 

7754 

78i6 

7879 

5o 

10 

9.564983 

5o48 

5ii3 

5i78 

5243 

53o8 

49 

IO 

9.587941 

8oo4 

8066 

8129 

8191 

8254 

49 

ii 

53y3 

5438 

55o3 

5568 

5633 

5698 

48 

ii 

83i6 

8379 

844i 

85o4 

8566 

8629 

48 

12 

5763 

5828 

5893 

5958 

6023 

6088 

47 

12 

8691 

8754 

8816 

8878 

894i 

9oo3 

47 

i3 

6x53 

6218 

6283 

6348 

64i3 

6478 

46 

i3 

9066 

9128 

9191 

9253 

93i5 

9378 

46 

i4 

654s 

6607 

6672 

6737 

6802 

6867 

45 

i4 

944o 

9502 

9565 

9627 

969o 

9752 

45 

i5 

6932 

6996 

7061 

7126 

7191 

7256 

44 

:5 

9814 

9877 

9939 

...  i 

63 

.  126 

44 

16 

7320 

7385 

745o 

75i5 

758o 

7644 

43 

16 

9.590188 

025o 

o3i3 

o375 

o437 

o499 

43 

i? 

7709 

7774 

7839 

79°3 

7968 

8o33 

42 

17 

o562 

0624 

0686 

o748 

0811 

o873 

42 

18 

8098 

8162 

8227 

8292 

8356 

8421 

4i 

18 

ogSS 

0997 

1060 

1122 

u84 

1246 

4i 

J9 

8486 

855o 

86i5 

8680 

8744 

8809 

4o 

J9 

i3o8 

1370 

!433 

1495 

i557 

i6i9 

4o 

20 

9.568873 

8938 

9003 

9067 

9l32 

9I97 

39 

20 

9.591681 

1743 

i8o5 

1868 

i93o 

1992 

39 

21 

9261 

9326 

9390 

9455 

95i9 

9584 

38 

21 

2o54 

2116 

2I78 

224O 

2302 

2364 

38 

22 

9648 

97i3 

9777 

9842 

99o6 

997i 

37 

22 

2426 

2488 

255o 

26l2 

2674 

2737 

37 

23 

9.570035 

OIOO 

0164 

0229 

0293 

o358 

36 

23 

2799 

2861 

2923 

2985 

3o47 

3io9 

36 

24 

0422 

0487 

o55i 

0616 

0680 

o744 

35 

24 

3i?I 

3232 

3294 

3356 

34i8 

348o 

35 

25 

0809 

0873 

o938 

IOO2 

1066 

n3i 

34 

25 

3542 

36o4 

3666 

3728 

379o 

3852 

34 

26 

1195 

I259 

I  324 

1  388 

i452 

i5i7 

33 

26 

3914 

3976 

4o38 

4099 

4i6i 

4223 

33 

27 

i58i 

1  645 

1710 

1774 

i838 

I9o3 

32 

27 

4285 

4347 

4409 

447i 

4532 

4594 

3a 

28 

1967 

2031 

2O95 

2160 

2224 

2288 

3i 

28 

4656 

4718 

478o 

4842 

49o3 

4965 

3i 

29 

2352 

2417 

248  1 

2545 

2609 

2673 

3o 

29 

5027 

5089 

5i5o 

5212 

5274 

5336 

3o 

3o 

9.572738 

2802 

2866 

293o 

2994 

3o59 

29 

3o 

9.595398 

5459 

552i 

5583 

5644 

57o6 

29 

3i 

3i23 

3187 

325i 

33i5 

3379 

3443 

28 

3i 

5768 

583o 

589i 

5953 

6oi5 

6o76 

25 

32 

3507 

357i 

3636 

3700 

3764 

3828 

27 

32 

6i38 

6200 

6261 

6323 

6385 

6446 

27 

33 

3892 

3956 

4020 

4o84 

4i48 

4212 

26 

33 

65o8 

6570 

663i 

6693 

6754 

6816 

26 

34 

4276 

434o 

44o4 

4468 

4532 

4596 

25 

34 

6878 

6939 

7ooi 

•7062 

7I24 

7i85 

25 

35 

466o 

4724 

4788 

4852 

49i6 

498o 

24 

35 

7247 

73o9 

737o 

7432 

7493 

7555 

24 

36 

5o44 

5io8 

5172 

5236 

5299 

5363 

23 

36 

7616 

7678 

7739 

78oi 

7862 

7924 

23 

37 

5427 

549i 

5555 

5619 

5683 

5747 

22 

37 

7985 

8o47 

8108 

8i7o 

823i 

8293 

22 

38 

58io 

5874 

5938 

6002 

6066 

6i3o 

21 

38 

8354 

84i5 

8477 

8538 

8600 

8661 

21 

39 

6193 

6257 

632i 

6385 

6449 

65i2 

20 

39 

8722 

8784 

8845 

8907 

8968 

9029 

2O 

4o 

9.576576 

664o 

6704 

6767 

683i 

6895 

J9 

4o 

9.599091 

9i52 

9213 

9275 

9336 

9398 

19 

4i 

6959 

7022 

7086 

7i5o 

72l3 

7277 

18 

4i 

9459 

9520 

958i 

9643 

97o4 

9765 

18 

42 

734i 

74o4 

7468 

7532 

7595 

7659 

J7 

42 

9827 

9888 

9949 

.  .11 

..72 

.133 

17 

43 

7723 

7786 

785o 

7914 

7977 

8o4i 

16 

43 

9.600194 

0256 

o3i7 

o378 

o439 

o5oo 

16  ' 

44 

8io4 

8168 

823i 

8295 

8359 

8422 

i5 

44 

o562 

0623 

0684 

o745 

0806 

0868 

i5  * 

45 

8486 

8549 

86i3 

8676 

874o 

88o3 

i4 

45 

0929 

0990 

io5i 

III2 

1174 

1235 

i4 

46 

8867 

893o 

8994 

9°57 

9I2I 

9i84 

i3 

46 

1296 

i357 

i4i8 

i479 

i54o 

1601 

iS 

47 

9248 

93n 

5375 

9438 

95o2 

9565 

12 

47 

i663 

I724 

i785 

1  846 

1907 

1968 

12 

48 

9629 

9692 

9755 

9819 

9882 

9946 

II 

48 

2029 

2090 

2l5l 

2212 

2273 

2334 

II 

49 

9.580009 

0072 

oi36 

0199 

0262 

o326 

IO 

49 

23g5 

2456 

25l7 

2578 

2639 

27OO 

10 

5o 

9.580389 

o453 

o5i6 

o579 

0642 

o7o6 

9 

5o 

9.602761 

2822 

2883 

2944 

3oo5 

3o66 

9 

5i 

0769 

o832 

0896 

o959 

1022 

1086 

8 

5i 

3127 

3i88 

3249 

33io 

3371 

3432 

52 

1149 

1212 

1275 

i339 

I4O2 

i465 

7 

52 

3493 

3554 

36i5 

3675 

3736 

3797 

7 

53 

i528 

iSgi 

i655 

1718 

1781 

1  844 

6 

53 

3858 

3919 

3980 

4o4i 

4lO2 

4162 

6 

54 

1907 

1971 

2034 

2097 

2l6o 

2223 

5 

54 

4223 

4284 

4345 

44o6 

4467 

4527 

5 

55 

2286 

235o 

24i3 

2476 

2539 

26O2 

4 

55 

4588 

4649 

47io 

477i 

483  1 

4892 

4 

56 

2665 

2728 

2791 

2854 

29I7 

298o 

3 

56 

4953 

5oi4 

5o74 

5i35 

5196 

5257 

o 
j 

57 

3o44 

3107 

3170 

3233 

3296 

3359 

2 

57 

53I? 

5378 

5439 

55oo 

556o 

562i 

2 

58 

3422 

3485 

3548 

36n 

3674 

3737 

I 

58 

5682 

5742 

58o3 

5864 

5924 

5985 

I 

59 

38oo 

3863 

3926 

3989 

4o52 

4n4 

o 

59 

6o46 

6106 

6i67 

6228 

6288 

6349 

O 

60" 

50" 

40" 

30" 

20" 

10" 

= 

60" 

50"  |  40" 

30" 

20" 

10" 

g 

Co-tangent  of  69  Degrees. 

~ 

Co-tangent  of  68  Degrees. 

p  P*  t$  *"  2"  3//  4//  5</  6//  7//  8//  9// 

p  T>  .11"  2"  3"  4"  5"  6"  7"  8"  9" 

J  G  13  19  26  32  39  45  51  58 

irl\  6   12  19  25  31  37  43  49  5(5 

LOGARITHMIC    SINES. 


.a 

Sine  of  22  Degrees. 

d 

Sine  of  23  Degrees. 

5 

0" 

10" 

20"  j  30"   40" 

50" 

* 

0" 

10" 

20" 

30" 

40" 

50" 

0 

I 

2 

2 

3888 
4200 
45ia 

3628 
394o 

4252 

4564 

3680^732  3784 
3992  4o44  4096 
43o4435644o8 
46i64668!472o 

3836 
4i48 
446o 

4772 

69 

58 

56 

O 
I 
2 

3 

9.59i878 
2176 

2770 

1928 

2225 
2522 
2819 

I977 
2275 
2572 

2869 

2027 

2324 

2621 

2918 

2076 
2374 
2671 
2968 

2126 

2423 

2720 
3017 

59 

58 

57 
56 

4 

4824 

4876:49284980 

5o32 

5o84 

55 

4 

3o67 

3n6 

3i65 

32i5 

3264 

33i4 

55 

5 

5i36 

518752395291 

5343 

5395 

54 

5 

3363 

3412 

3462 

35n 

356i 

36io 

54 

6 

5447 

5499  555o  56o2 

5654 

57o6 

53 

6 

3659 

3709 

3758 

38o7 

3857 

3906 

53 

7 

5758 

58io586i  5913 

5965 

6017 

52 

7 

3955 

4oo5 

4o54 

4io3 

4i53 

4202 

52 

8 

6069 

6120  6172,6224 

62-76 

6327 

5i 

8 

425i 

43oi 

435o 

4399 

4448 

4498 

5i 

9 

6379 

643  1 

6482  6534 

6586 

6638 

5o 

9 

4547 

4596 

4645 

4695 

4744 

4793 

5o 

IO 

9.576689 

674i 

6793  6844 

6896 

6948 

49 

10 

9.594842 

4891 

494i 

499o 

5o88 

49 

ii 

6999 

7o5i 

7io2  7i54 

-7206 

7257 

48 

ii 

5i37 

5i86 

5236 

5285 

533; 

5383 

48 

12 

73o9 

7360  74i2  7464 

75i5 

7567 

47 

12 

5432 

548! 

553o 

558o 

5629 

5678 

47 

i3 

7618 

767o 

772i  7773 

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46 

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5825 

5874 

592^ 

5972 

46 

i4 

7927 

7979 

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8i33 

8i85 

45 

i4 

6021 

6070 

6119 

6168 

6217 

6266 

45 

i5 

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82888339839i 

8442 

8494 

44 

i5 

63i5 

6364 

64i3 

6462 

65n 

656o 

44 

16 

8545 

8596  8648,8699  875i 

8802 

43 

16 

6609 

6658 

6707 

6756 

68o5 

6854 

43 

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8853 

8956!9oo8jco59 

9TIO 

42 

17 

69o3 

6952 

7001 

7o5o 

7099 

7i48 

42 

18 

9162 

9213  9264  9316  o367 

4i 

18 

7i96 

7245 

7294 

7343 

744i 

4i 

i9 

9470 

952I 

95^9623 

9675 

9726 

4o 

19 

749o 

7539 

7587 

7636 

7685 

7734 

4o 

20 

9.579777 

9880  9931 

9982 

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39 

20 

9.597783 

783i 

7880 

7929 

7978 

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39 

21 

9.58oo85 

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0238 

0289 

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38 

21 

8o75 

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38 

22 

0392 

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37 

22 

8368 

84i7 

8465 

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8612 

37 

23 

0699 

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36 

23 

8660 

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36 

24 

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1209 

1261 

35 

24 

8952 

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35 

25 

l3l2 

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i5i6 

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34 

25 

9244 

9293 

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34 

26 

1618 

1669 

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1771 

1822 

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33 

26 

9536 

9584 

9633 

9681 

973o 

9778 

33 

27 

1924 

1975 

2025 

2076 

2I27 

2178 

32 

27 

9827 

9876 

9924 

9973 

.  .21 

..7o 

32 

28 

2229 

2280  233i 

2382 

2433 

2484 

3i 

28 

9.  600118 

0167 

02l5 

0264 

0312 

o36i 

3i  , 

29 

2535 

2585 

2636 

2687 

2738 

2789 

3o 

2.9    0409 

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3o 

3o 

9,582840 

2890 

2941 

2992 

3o43 

3094 

29 

3o  9.600700 

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0942 

29 

3i 

3i45 

3i95 

3246 

3297 

3348 

3398 

28 

3i 

o99o 

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1087 

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1184 

1232 

28 

32 

3449 

35oo 

355i 

36oi 

3652 

37o3 

27 

32 

1280 

1329 

i377 

i425 

i474 

l522 

27 

33 

3754 

38o4 

3855 

3906 

3956 

4007 

26 

33 

i57o 

1619 

1667 

1716 

1763 

1812 

26 

34 

4o58 

4io8 

4210 

4260 

43n 

25 

34 

1860 

1908 

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2o53 

2101 

25 

35 

436i 

44i2 

4463 

45i3 

4564 

46i5 

24 

35 

2i5o 

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2246 

2294 

2342 

239I 

24 

36 

4665 

4716^4766 

4817 

4867 

4918 

23 

36 

2439 

2487 

2535 

2583 

2632 

2680 

23 

37 

4968 

5oi9 

5o7o 

5l2O 

6171 

5221 

22 

37 

2728 

2776 

2824 

2872 

2920 

2969 

22 

38 

5272 

5322 

5373 

5423 

5474 

5524 

21 

38 

3oi7 

3o65 

3n3 

3i6i 

3209 

3257 

21 

39 

5574 

5625 

5675 

5726 

5776 

5827 

20 

39 

33o5 

3353 

34oi 

3449 

3497 

3546 

2O 

4o 

9.  585877 

5927  5978 

6028 

6o79 

6129 

T9 

4o 

9.6o3594 

3642 

3690 

3738 

3786 

3834 

1  9 

4i 

6179 

623o  6280 

633i 

638i  643i 

18 

4i 

3882 

393o 

3978 

4026 

4074 

4l22 

18 

42 

6482 

6532 

6582 

6633 

6683  6733 

17 

42 

4i7o 

4218 

42G6 

43i3 

436i 

4409 

i7 

43 

6783 

68346884 

6934 

6985  7o35 

16 

43 

4457 

45o5 

4553 

46oi 

4649 

4697 

16 

44 

7o85 

7i35 

7i86 

7236 

-7286  7336 

i5 

44 

4745 

4793 

484i 

4888 

4936 

4984 

i5 

45 

7386 

7437 

7487 

7537 

75877637 

i4 

45 

5o32 

5o8o 

5i28 

5176 

5223 

527I 

i4 

46 

7688 

7738 

7788 

7838 

-7888  7938 

i3 

46 

53i9 

5367 

54i5 

5462 

55io 

5558 

i3 

47 

7989 

8039 

8089 

8i39 

81898239 

12 

47 

56o6 

5654 

5701 

5749 

5797 

5845 

12 

48 

8289 

83398389 

8439 

8489  854o 

II 

48 

5892 

5940 

5988 

6o35 

6o83 

6i3i 

II 

49 

8590 

8690 

874o 

879o  884o 

IO 

49 

6i79 

6226 

6274 

6322 

6369 

64i7 

IO 

5o 

9.588890 

8940  8990 

9o4o 

9090  9140 

9 

5o 

9.6o6465 

65i2 

656o 

6608 

6655 

67o3 

9 

5i 

9190 

92400290 

934o 

9389  9439 

8 

5i 

675i 

6798 

6846 

6893 

694i 

6989 

8 

52 

9489 

95399689 

9639 

9689  9739 

7 

52 

7o36 

7o84 

7i3i 

7179 

7227 

7274 

7 

53 

9789 

9839' 

9889 

9938 

9988  ..38 

6 

53 

7322 

7369 

7417 

7464 

7512 

7559 

6 

54 

9  ,590088 

oi38' 

0188 

0237 

028-7  °337 

5 

54 

7607 

7654 

7702 

7749 

7797 

7844 

5 

55 

0387 

o437 

0487 

o536 

o586o636 

4 

55 

7892 

7939 

7987 

8o34 

8082 

8129 

4 

56 
57 

0686 
0984 

o735 
io34 

o785 
1084 

o835 
n33 

08850934 
n83  1233 

3 

2 

56 

57 

8177 

846i 

8224 
85o8 

8271 
8556 

8319 
86o3 

8366 
865  1 

8698 

3 

2 

58 

1282 

i332 

i382 

i43i 

i48i 

i53i 

I 

58 

8745 

8793 

884o 

8887 

8935 

8982 

1  ! 

69 

i58o 

i63o 

1680 

1-729 

i779  1828 

O 

59 

9029 

9077 

9124 

9171 

9219 

9266  o 

60" 

50" 

40"   30"  I  20"  1  10" 

e 

60" 

50"   40"   30" 

20" 

TO" 

d 

Co-sine  of  67  Degrees. 

I 

Co-sine  of  66  Degrees.     % 

t  ^11  o"  3' 

A"  c//  KII  7;r  on  Q// 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

P.   Part  J      g          1Q        J.        2Q       ^g        -j       gg        ~J       ~g 

P'iart)  5  10  15  1.9  24  29  34  39  44 

LOGARITHMIC    TANGENTS. 


J 

Tangent  of  22  Degrees. 

| 

Tangent  of  23  Degrees. 

s 

0" 

10"   20" 

30" 

40"   50" 

* 

0" 

10" 

20" 

30" 

40"  |  50" 

o 

9.  6o(?4io 

6470 

653i 

659i 

6652 

67i3 

59 

o 

9.627852 

•7910 

7969 

8028 

8086 

8i45 

59 

I 

6773 

6834 

6894 

6955 

7oi5 

7076 

58 

I 

8203 

8262 

8320 

8379 

8437 

8496 

58 

2 

7i37 

7197 

7258 

73i8 

7379 

7439 

57 

2 

8554 

8612 

8671 

8729 

8788 

8846 

57 

3 

75oo 

756o 

7621 

7681 

7742 

7802 

56 

3 

8905 

8963 

9022 

9080 

9i38 

9197 

56 

4 

7863 

7923 

7984 

8o44 

8io5 

8i65 

55 

4 

9255 

93i4 

9372 

943  1 

9489 

9547 

55 

5!    8225 

8286 

8346 

84o7 

8467 

8528 

54 

5 

9606 

9664 

9722 

9781 

9839 

9897 

54 

6    8588 

8648 

8709 

8769 

883o 

8890 

53 

6 

9956 

.  .  i4 

..73 

.  i3i 

.i89 

.247 

53 

7 

8950 

9011 

9071 

9i3i 

9192 

9252 

52 

7 

9.63o3o6 

o364 

0422 

o48i 

o539 

o597 

5a 

8 

9312 

9373 

9433 

9493 

9554 

9614 

5i 

8 

o656 

o7i4 

0772 

o83o 

0889 

o947 

5i 

9 

9674 

9735 

9795 

9855 

99i5 

9976 

5o 

9 

ioo5 

io63 

1122 

1180 

1238 

I296 

5o 

10 

9.6ioo36 

0096 

oi56 

0217 

0277 

o337 

49 

IO 

9.63i355 

i4i3 

1471 

1529 

1587 

1  646 

49 

ii 

o397 

o458 

o5i8 

o578 

o638 

0698 

48 

ii 

1704 

I762 

1820 

1878 

1936 

i995 

48 

12 

°759 

0819 

0879 

o939 

°999 

1069 

47 

12 

2o53 

2III 

2169 

2227 

2285 

2343 

47 

i3 

II2O 

1180 

1240 

i3oo 

i36o 

1420 

46 

i3 

2402 

2460 

25i8 

2576 

2634 

2692 

46 

i4 

i48o 

i54o 

1601 

1661 

1721 

1781 

45 

i4 

275o 

2808 

2866 

2924 

2982 

3o4o 

45 

i5 

i84i 

1901 

1961 

2021 

2081 

2l4l 

44 

i5 

3099 

3i57 

32i5 

3273 

333i 

3389 

44 

16 

22OI 

2261 

2321 

238i 

244i 

25oi 

43 

16 

3447 

35o5 

3563 

362i 

3679 

3737 

43 

17 

256i 

2621 

2681 

2741 

2801 

2861 

42 

17 

3795 

3853 

3911 

3969 

4027 

4o85 

42 

18 

292I 

2981 

3o4i 

3ioi 

3i6i 

3221 

4i 

18 

4i43 

4201 

4259 

43i6 

4374 

4432 

4i 

19 

328l 

334i 

34oi 

346i 

352i 

358i 

4o 

I9 

4490 

4548 

46o6 

4664 

4722 

4780 

4o 

20 

9.6i364i 

37oi 

376o 

3820 

388o 

3940 

39 

20 

9.634838 

4896 

4954 

Son 

5069 

5127 

39 

21 

4ooo 

4o6o 

4l2O 

4i8o 

4239 

4299 

38 

21 

5i85 

5243 

53oi 

5359 

54i6 

5474 

38 

22 

4359 

44i9 

4479 

4539 

4598 

4658 

37 

22 

5532 

5590 

5648 

5706 

5763 

582i 

37 

23 

4718 

4778 

4838 

4897 

4957 

5017 

36 

23 

5879 

5937 

5995 

6o52 

6110 

6168 

36 

24 

5077 

5i36 

5i96 

5256 

53i6 

5375 

35 

24 

6226 

6283 

634i 

6399 

6457 

65i4 

35 

25 

5435 

5495 

5555 

56i4 

5674 

5734 

34 

25 

6572 

663o 

6688 

6745 

68o3 

6861 

34 

26 

5793 

5853 

5913 

5972 

6o32 

6092 

33 

26 

6919 

6976 

7o34 

7092 

7149 

7207 

33 

27 

6i5i 

6211 

6271 

633o 

6390 

645o 

32 

27 

7265 

7322 

738o 

7438 

7495 

7553 

32 

28 

6509 

6569 

6628 

6688 

6748 

6807 

3i 

28 

7611 

7668 

7726 

7783 

7841 

7899 

3i 

29 

6867 

6926 

6986 

7046 

7io5 

7i65 

3o 

29 

7956 

8014 

8072 

8i29 

8187 

8244 

3o 

3o 

9  617224 

7284 

7343 

74o3 

7462 

7522 

29 

3o 

9.  638302 

8359 

84i7 

8475 

8532 

SSgo 

29 

3i 

7582 

7641 

7701 

7760 

7820 

7879 

28 

3i 

8647 

8705 

8762 

8820 

8877 

8935 

28 

32 

7939 

7998 

8057 

8117 

8176 

8236 

27 

32 

8992 

9o5o 

9107 

9i65 

9222 

9280 

27 

33 

8295 

8355 

84i4 

8474 

8533 

8593 

26 

33 

9337 

9395 

9452 

95io 

9567 

9625 

26 

34 

8652 

8711 

8771 

883o 

8890 

8949 

25 

34 

9682 

974o 

9797 

9855 

9912 

9969 

25 

35 

9008 

9068 

9127 

9186 

9246 

93o5 

24 

35 

9.640027 

0084 

0142 

0199 

0257 

o3i4 

24 

36 

9364 

9424 

9543 

9602 

9661 

23 

36 

0371 

0429 

o486 

o544 

0601 

o658 

23 

37 

9720 

9780 

9839 

9898 

9958 

..17 

22 

37 

0716 

0773 

o83o 

0888 

o945 

IOO2 

22 

38 

9.620076 

oi36 

0195 

0254 

o3i3 

o373 

21 

38 

1060 

1174 

1232 

1289 

1  346 

21 

39 

o432 

0491 

o55o 

0610 

0669 

0728 

20 

39 

i4o4 

i46i 

i5i8 

i575 

i633 

1690 

2O 

4o 

9.620787 

o846 

0906 

o965 

1024 

io83 

19 

4o 

9.641747 

i8o5 

1862 

1919 

1976 

2034 

19 

4i 

1142 

I2OI 

1261 

1320 

i379 

i438 

18 

4i 

2091 

2148 

22O5 

2263 

2320 

2377 

18 

42 

1497 

i556 

1616 

i675 

1734 

i793 

17 

42 

2434 

249I 

2549 

2606 

2663 

272O 

17 

43 

i852 

1911 

1970 

2029 

2088 

2147 

16 

43 

2777 

2834 

2892 

2949 

3oo6 

3o63 

16 

44 

2207 

2266 

2325 

2384 

2443 

25O2 

i5 

44 

3l20 

3i77 

3235 

3292 

3349 

34o6 

i5 

45 

256i 

2620 

2679 

2738 

2797 

2856 

i4 

45 

3463 

3520 

3577 

3634 

369i 

3749 

i4 

46 

2915 

2974 

3o33 

3092 

3i5i 

3210 

i3 

46 

38o6 

3863 

3920 

3977 

4o34 

4091 

i3 

47 

3269 

3328 

3387 

3446 

35o5 

3564 

1  2 

47 

4i48 

42o5 

4262 

43i9 

4376 

4433 

12 

48 

3623 

3682 

374i 

38oo 

3858 

39i7 

II 

48 

4490 

4547 

46o4 

466  1 

4718 

4775 

I  I 

49 

3976 

4o35 

4094 

4i53 

4212 

4271 

IO 

49 

4832 

4889 

4946 

5oo3 

5o6o 

IO 

5o 

9.62433o 

4388 

4447 

45o6 

4565 

4624 

o 

5o 

9.64517*4 

523i 

5288 

5345 

5402 

5459 

9 

5i 

4683 

4742 

4800 

4859 

4918 

4977 

8 

5i 

55i6 

5573 

563o 

5687 

5744 

58oi 

8 

52 

5o36 

5og4 

5i53 

5212 

5271 

533o 

7 

52 

5857 

59i4 

597i 

6028 

6o85  6142 

7 

53 

5388 

5447 

55o6 

5565 

5623 

5682 

6 

53 

6199 

6256 

63i3 

6369 

6426  6483 

6 

54 

5741 

58oo 

5858 

59i7 

5976 

6o35 

5 

54 

654o 

6597 

6654 

6710 

67676824 

5 

55 

6093 

6i52 

6211 

6269 

6328 

6387 

4 

55 

6881 

6938 

6995 

7o5i 

7108  7165 

4 

56 

6445 

65o4 

6563 

6621 

6680 

6739 

3 

56 

7222 

7279 

7335 

7392 

7449  75o6 

3 

57 

6797 

6856 

6915 

6973 

7032 

7090 

2 

57 

7562 

7619 

7676 

7733 

7789  7846 

2 

58 

7i49 

7208 

7266 

7325 

7383 

7442 

I 

58 

79°3 

7960  8016 

8073 

8i3o8i86 

I 

59 

7559 

7618 

7676 

7735 

7793 

0 

59 

8243 

83oo 

8356 

84i3 

84708526 

O 

60" 

50"  |  40" 

30"   20" 

10" 

d 

60"     50" 

40"   30"  1  20"  1  10" 

Co-tangent  of  67  Degrees. 

.H 

Co-tangent  of  66  Degrees. 

1 

p  p   j  1"  2"  3"  4"  5"  6"  7"  8"  9" 

p  p   (  1"  2"  3"  4"  5"  6"  7"  8"  9;/ 

I  6  12  18  24  30  36  42  48  54 

"*{  6  12  17  23  29  35  40  46  52 

LOGARITHMIC    SINES. 


IT 

Sine  of  24  Degrees. 

g-      Sine  of  25  Degrees. 

% 

0' 

10" 

20" 

30" 

40" 

50" 

i 

0" 

10" 

20" 

30" 

40'' 

50" 

0 

9  609313 

936i 

94o8 

9455 

95o2 

955o 

59 

o 

9.625948 

6993 

6o39 

6o84 

6l29 

6i74 

59 

i 

9597 

9644 

9691 

9739 

9786 

9833 

58 

i 

62I9 

6264 

63o9 

6354 

64oo 

6445 

58 

2 

9880 

9928 

9975 

.  .22 

..69 

.116 

57 

2 

649o 

6535 

658o 

6625 

667o 

67i5 

5? 

3 

9  ,6ioi64 

02  1  1 

0258 

o3o5 

o352 

o399 

56 

3 

6760 

68o5 

685o 

68^5 

694o 

6985 

56 

4 

o44y 

0494 

o54i 

o588 

o636 

0682 

55 

4 

.  7o3o 

7075 

7120 

7i65 

72IO 

7255 

55 

5 

0729 

0776 

0823 

0871 

o9i8 

o965 

54 

5 

73oo 

7345 

739° 

7435 

748o 

7525 

54 

6 

IOI2 

io59 

1106 

n53 

I2OO 

1247 

53 

6 

757° 

76i5 

7660 

7705 

775o 

7795 

53 

7 

1294 

i34i 

i388 

i435 

1482 

i529 

52 

7 

784o 

7885 

7929 

7974 

8oi9 

8o64 

52 

8 

1576 

1623 

1670 

1717 

1764 

1811 

5i 

8 

8io9 

8i54 

8i99 

8244 

8289 

8333 

5i 

9 

i858 

1905 

1952 

:999 

2046 

2O93 

5o 

9 

8378 

8423 

8468 

85i3 

8558 

8602 

5o 

10 

9.612140 

2187 

2234  2280 

2327 

2374 

49 

10 

9.  628647 

8692 

8737 

8782 

8826 

887i 

49 

ii 

2421 

2468 

25i5  2562 

26o9 

2655 

48 

ii 

89i6 

896i 

9oo6 

9o5o 

9o95 

9i4o 

48 

12 

2702 

2749 

2796^843 

289o 

2936 

47 

12 

9i85 

9229 

9274 

93i9 

9363 

94o8 

47 

i3 

2983 

3o3o 

30773124 

3171 

3217 

46 

i3 

9453 

9498 

9542 

9587 

9632 

9676 

46 

i4 

3264 

33n 

3358  34o4 

345  1 

3498 

45 

i4 

9721 

9766 

98io 

9855 

99oo 

9944 

45 

r5 

3545 

359i 

3638  3685 

3732 

3778 

44 

i5 

9989 

..34 

..78 

.123 

.168 

.212 

44 

16 

3825 

3872 

39i83965 

4OI2 

4o58 

43 

16 

9.630257 

o3oi 

o346 

o39i 

o435 

o48o 

43 

17 

4io5 

4i52 

4i984245 

4292 

4338 

42 

J7 

o524 

o569 

o6i3 

o658 

o7o3 

o747 

42 

18 

4385 

4432 

44784525 

457i 

46i8 

4i 

18 

0792 

o836 

0881 

0925 

o970 

1014 

4i 

'9 

4665 

4711 

475848o4 

485i 

4898 

4o 

T9 

1059 

no3 

n48 

II92 

1237 

1281 

4o 

20 

9.614944 

4991 

5o375o84 

5i3o 

5177 

39 

20 

9.63i326 

1370 

i4i5 

i459 

i5o4 

1  548 

39 

21 

5223 

5270 

53i65363 

54o9 

5456 

38 

21 

i593 

i637 

1681 

1726 

1770 

i8i5 

38 

22 

55o2 

5549 

5595  5642 

5688 

5735 

37 

22 

1859 

I9o4 

i948 

I992 

2037 

2081 

37 

23 

578i 

5828 

5874  592i 

5967 

60  1  3 

36 

23 

2125 

2170 

22l4 

2259 

23o3 

2347 

36 

24 

6060 

6106 

6i53  6i99 

6245 

6292 

35 

24 

2392 

2436 

2480 

2525 

2569 

26i3 

35 

25 

6338 

6385 

643i  6477 

6524 

657O 

34 

25 

2658 

2702 

2746 

2790 

2835 

2879 

34 

26 

6616 

6663 

67O9  6755 

6802 

6848 

33 

26 

2923 

2968 

3OI2 

3o56 

3ioo 

3i45 

33 

27 

6894 

6941 

6987  7033 

7080 

7126 

32 

27 

3i89 

3233 

3277 

3322 

3366 

34io 

32 

28 

.  71?2 

7218 

7265  73n 

7357 

74o3 

3i 

28 

3454 

3498 

3543 

3587 

363i 

3675 

3i 

29 

745o 

7496 

7542  7588 

7635 

7681 

3o 

29 

37i9 

3?64 

38o8 

3852 

3896 

394o 

3o 

3o 

9.617727 

7773 

78i9  7866 

79I2 

79.58 

29 

3o 

9.633984 

4028 

4o73 

4117 

4i6i 

42O5J29 

3i 

8oo4 

8o5o 

8o968i43 

8i89 

8235 

28 

3i 

•4249 

4293 

4337 

438i 

4426 

4470 

28 

32 

8281 

8327 

837384i9 

8465 

85I2 

27 

32 

45i4 

4558 

4602 

4646 

469o 

4734 

27 

33 

8558 

86o4 

865o8696 

8742 

8788 

26 

33 

4778 

4822 

4866 

49io 

4954 

4998 

26 

34 

8834 

8880 

8926  8972 

9oi8 

9o64 

25 

34 

5o42 

5o86 

5i3o 

5i74 

5218 

5202 

25 

35 

9110 

9i56 

92O2  9248 

9294 

934o 

24 

35 

53o6 

535o 

5394 

5438 

5482 

5526 

24 

36 

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36 

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2728 

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49 

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10 

49 

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62 

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52 

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53 

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6 

53 

9.  640024 

0068 

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6 

54 

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4455 

45oo 

4546 

5 

54 

0284 

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5 

55 

459i 

4636 

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4 

55 

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4 

56 

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4954 

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3 

56 

0804 

0848 

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o934 

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3 

5? 

5i35 

5i8o 

5225 

5270 

53i6 

536i 

2 

57 

1064 

1107 

n5i 

n94 

I237 

1280 

2 

58 

54o6 

545  1 

5496 

5542 

5587 

5632 

I 

58 

1324 

i367 

i4io 

i453 

1496 

i54o 

I* 

59 

5677 

5722 

5768 

58i3 

5858 

5903 

O 

59 

i583 

1626 

i669 

I7I2 

17^6 

-799  °  . 

60" 

50" 

40" 

30"   20" 

10" 

c 

60"    |  50"   40" 

30"  |  20"  |  10" 

d 

Co-sine  of  65  Degrees. 

S 

Co-sine  of  Q4  Degrees. 

1 

1 

„  „  ,<  1"  2"  3"  4"  5"  6"  7"  8"  9" 
in\  5   9  14  18  23  28  32  37  42 

.  (  1"  2"  3"  4"  5"  6"  7"  8"  9"  \ 
irt\  4   9  13  18  22  26  31  35  40 

LOGARITHMIC    T  A  N  i;  K  x  T  s. 


J3 

Tangent  of  24  Degrees. 

.g 

Tangent  of  25  Degrees. 

9 

0" 

10" 

20" 

30" 

40" 

50" 

^ 

0" 

10" 

[20" 

30" 

40" 

50" 

o 

9.648583 

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9.668673 

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5421 

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62 

6029 

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6i39 

6i94 

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7 

52 

56i2 

5666 

57i9 

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7 

53 

636o 

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6525 

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6 

53 

5934 

5987 

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6i48 

6202 

6 

54 

6691 

6746 

6801 

6856 

69i  i 

6966 

5 

54 

6255 

63o9 

6363 

64i6 

6470 

6523 

5 

55 

7021 

7076 

7l32 

7187 

7242 

7297 

4 

55 

6577 

663o 

6684 

6737 

679i 

6845 

4 

56 

7352 

7407 

7462 

75i7 

7572 

7627 

3 

56 

6898 

6952 

7oo5 

7°59 

7112 

7166 

3 

5n 

7682 

7737 

7792 

7847 

79o3 

7958 

2 

57 

7219 

7273 

7326 

738o 

7433 

7487 

2 

58 

8oi3 

8068 

8i23 

8i78 

8233 

8288 

I 

58 

754o 

7594 

7647 

7701 

7754 

7808 

I 

59 

8343 

8398 

8453 

85o8 

8563 

8618 

0 

59 

7861 

79i5 

7968 

8021 

8o75 

8128 

O 

60" 

50"  j  40" 

30"   20" 

10" 

g 

60" 

50"  |  40" 

30"   20"  |  10" 

Co-tangent  of  65  Degrees. 

2, 

Co-tangent  of  64  Degrees. 

§ 

p  p   <1"  2"  3"  4"  5"  6"  7"  8"  9" 
)  (5  11  17  22  28  33  39  45  50 

!         '                                 ' 

p  p  A  I"  2"  3"  4"  5"  6"  7"  8"  9" 
ir  \  5   11  16  22  27  33  38  43  49 

D 


50 


LOGARITHMIC    SINES. 


1 

Sine  of  26  D  grees. 

.3 

Sine  of  27  Degrees. 

m 

0" 

10" 

20"   1  0"   40" 

.V 

rt 

0" 

10* 

20"  1  30" 

40" 

50" 

o 

9.641842 

i885 

I928 

I97I  20l5 

21'"  8 

59 

0 

9.  657047 

7o88 

7I29 

7171 

72I2 

7253 

59 

i 

2101 

2i44 

2l87 

22302273 

23  "7 

58 

i 

7295 

7336 

7377 

74i8 

746o 

75oi 

58 

2 

236o 

24o3 

2446 

2489 

2532 

25'rS 

57 

2 

7542 

7584 

7625 

7666 

77°7 

7749 

57 

3 

2618 

2661 

2704 

2747 

2790 

28}  \ 

56 

3 

779° 

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79i3 

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7996 

56 

4 

2877 

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4 

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8243 

55 

5 

3i35 

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3264 

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54 

5 

8284 

8325 

8367 

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54 

6 

3393 

3436 

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3522 

3565 

36o7 

53 

6 

853i 

8572 

86i3 

8655 

8696 

8737 

53 

7 

365o 

3693 

3736 

3779 

3822 

3865 

52 

7 

8778 

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8860 

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8942 

8983 

52 

8 

3908 

395i 

3994 

4o37 

4o8o 

4123 

5i 

8 

9O25 

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9i89 

923o 

5i 

9 

4i65 

4208 

4s5i 

4294 

4337 

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5o 

9 

9271 

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10 

ii 

9.644423 
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4808 

4594 
485i 

4637 
4894 

49 

48 

10 
ii 

9.6595i7 
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9558 
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964o 
9886 

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9927 

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49 
48 

12 

4936 

4979 

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47 

12 

9.66ooo9 

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43 

17 

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20 

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39 

20 

9.  661970 

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493i 

4971 

5OI2 

5o52 

5oQ3 

27 

33 

0287 

0829  o37i 

o4i3 

0455 

0497 

26 

33 

5i33 

5i73 

5254 

5294 

5335 

26 

34 

o539 

o582  0624 

0666 

0708 

0750 

25 

34 

5375 

54i5 

5456 

5496 

5536 

5577 

25 

35 

O792 

o834o876 

o9i8 

o96o 

1  002 

24 

35 

56i7 

5657 

5697 

5738 

5778 

58i8 

24 

36 

1044 

1086 

1128 

n7i 

I2l3 

1255 

23 

36 

5859 

5899 

5939 

5979 

6020 

6060 

23 

37 

I297 

i339 

i38i 

i465 

1507 

22 

37 

6100 

6i4o 

6181 

6221 

6261 

63oi 

22 

38 

i549 

i59i 

i633 

i675 

1716 

i758 

21 

38 

6342 

6382 

6422 

6462 

65o2 

6543 

21 

39 

1800 

1842 

i884 

I926 

i968 

2OIO 

2O 

39 

6583 

6623 

6663 

67o3 

6743 

6784 

2O 

4o 

Q  •  652O32 

2094 

2i36 

2I78 

2220 

2262 

I9 

4o 

9.666824 

6864 

69o4 

6944 

6984 

7O25 

I9 

4i 

23o4 

2345 

2387 

2429 

2471 

25i3 

18 

4i 

7o65 

7io5 

7i45 

7i85 

7225 

7265 

18 

42 

2555 

2597 

2638 

2680 

2722 

2764 

17 

42 

73o5 

7346 

7386 

7426 

7466 

75o6 

i7 

43 

2806 

2848 

289O 

293i 

2973 

3oi5 

16 

43 

7546 

7586 

7626 

7666 

7706 

7746 

16 

44 

3o57 

3o99 

3i4o 

3182 

3224 

3266 

i5 

44 

7786 

-7826 

7866 

79o6 

7946 

7986 

i5 

45 

33o8 

3349 

339i 

3433 

3475 

35i6 

i4 

45 

8027 

8o67 

8107 

8i47 

8187 

822-7 

i4 

46 

3558 

36oo  3642 

3683 

3725 

3767 

i3 

46 

8267 

83o7 

8347 

8386 

8426 

8466 

i3 

47 

38o8 

385o3892 

3934 

3975 

4017 

12 

47 

85o6 

8546 

8586 

8626 

8666 

87o6 

12 

48 
49 

4o59 
43o9 

4ioo  4i42 
435o4392 

4i84 
4434 

4225  4267 
4475  U5  1  7 

II 
10 

48 
49 

8746 
8986 

8786 

9O26 

8826 
9o65 

8866 
9io5 

89o6 
9i45 

8946 
9i85 

II 
IO 

5o 

9.  654558 

4600  4642 

4683 

4725 

4766 

9 

5o 

9.669225 

9265 

93o5 

9345 

9384 

9424 

9 

5i 

48o8 

485o489i 

4933 

4974 

5oi6 

8 

5i 

9464 

95o4 

9544 

9584 

9624 

9663 

8 

52 

5o58 

5o99  5i4i 

5i82 

5224  5265 

7 

52 

97°3 

9743 

9783 

9823 

9862 

9902 

7 

53 

53o7 

5348 

53oo 

543  1 

547355i4 

6 

53 

9942 

9982 

.  .22 

..61 

.  101 

.  i4i 

6 

54 

5556 

55975639 

568o 

5722  5763 

5 

54 

9.67oi8i 

O22O 

O260 

o3oo 

o34o 

o379 

5 

55 

58o5 

5846 

5888 

5929 

597i  6012 

4 

55 

o4i9 

o459 

o499 

o538 

o578 

0618 

4 

56    6o54 

6o95 

6i36 

6i78 

62i9  6261 

3 

56 

o658 

o697 

o737 

0777 

0816 

o856 

3 

57 

63o2 

6344 

6385 

6426 

6468  65o9 

2 

57 

o896 

o935 

o975 

ioi5 

io54 

io94 

2 

58 

655i 

6592 

6633 

6675 

6716  6757 

I 

58 

n34 

n73 

I2l3 

1253 

I292 

i332 

I 

59 

6799 

684o  6881 

6923 

6964  7oo5 

0 

59 

l372 

i4n  i45i 

i49o 

i53o 

i57o 

0 

60" 

50" 

40"   30" 

20"   10" 

B 

60" 

50"  |  40" 

30" 

20" 

10" 

Co-sine  of  63  Degrees. 

.B 

Co-sine  of  62  Degrees. 

| 

p  p   <  1"  2"  3"'  4"  5"  6"  7"  8'  9" 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

l\  4   8  13  17  21  25  30  34  38 

}  4   8  12  16  20  24  28  32  36  j 

LOGARITHMIC    TANGENTS. 


c 

Tangent  of  26  Degrees. 

j 

Tangent  of  27  Degrees. 

i 

0"    |  10"  |  20"   30"   40"  j  50" 

s 

0" 

10"  |  20" 

30" 

40" 

50" 

0 

9.688182 

8235 

8289 

8342 

8395  844g 

59 

o 

9.707166 

72l8 

7270 

7322 

7374 

7426 

59 

I 

85o2 

8556 

8609 

8663 

8716 

8769 

58 

i 

7478 

753o 

7582 

7634 

7686 

7738 

58 

2 

8823 

8876 

8930 

8983 

9036 

9090 

57 

2 

779° 

7842 

7894 

7946 

7998 

8o5o 

57 

3 

9i43 

9196 

925o 

93o3 

9356 

94lO 

56 

3 

8102 

8i54 

8206 

8258 

83io 

8362 

56 

4 

9463 

95i6 

957° 

9623 

9676 

9730 

55 

4 

84i4 

8466 

85i8 

857o 

8622 

8674 

55 

5 

9783 

9836 

9890 

9943 

9996 

:.5o 

54 

5 

8726 

8778 

883o 

8382 

8934 

8985 

54 

6 

9.690103 

oi56 

O2IO 

0263 

o3i6 

0369 

53 

6 

9o37 

9089 

9141 

9i93 

9245 

9297 

53 

7 

0423 

0476 

0529 

o582 

o636 

0689 

52 

7 

9349 

9401 

9453 

95o4 

9556 

96o8 

52 

8 

0742 

o795 

0849 

0902 

o955 

1008 

5i 

8 

966o 

9712 

9764 

9816 

9868 

99i9 

5i 

9 

1062 

ni5 

1168 

1221 

1274 

i328 

5o 

9 

9971 

..23 

..75 

.12-7 

.179 

.231 

5o 

10 

9.69i38i 

i434 

i487 

i54o 

i594 

1647 

49 

IO 

9.  710282 

o334 

o386 

o438 

o49o 

o542 

49 

ii 

1700 

i753 

1806 

i859 

1913 

1966 

48 

ii 

o593 

o645 

0697 

o749 

0801 

o852 

48 

12 

2019 

2072 

2125 

2178 

2232 

2285 

47 

12 

o9o4 

o956 

1008 

io59 

mi 

n63 

47 

i3 

2338 

2391 

2444 

2497 

255o 

26o3 

46 

i3 

I2l5 

1267 

i3i8 

i37o 

1422 

i474 

46 

i4 

2656 

2710 

2763 

2816 

2869 

2922 

45 

i4 

i525 

i577 

1629 

1681 

1732 

i784 

45 

i5 

2975 

3028 

3o8i 

3i34 

3i87 

324o 

44 

i5 

i836 

1887 

i939 

1991 

2o43 

2094 

44 

16 

3293 

3346 

34oo 

3453 

35o6 

3559 

43 

16 

2  1  46 

2198 

2249 

2301 

2353 

24o5 

43 

i? 

36i2 

3665 

37i8 

377i 

3824 

3877 

42 

17 

2456 

25o8 

256o 

2611 

2663 

27l5 

42 

18 

SgSo 

3983 

4o36 

4089 

4142 

4i95 

4i 

18 

2766 

28!8 

2870 

2921 

2973 

3o25 

4i 

19 

4248 

43oi 

4354 

4407 

446o 

45i3 

4o 

r9 

3o76 

3i28 

3i79 

323i 

3283 

3334 

4o 

20 

9.694566 

4619 

4672 

4724 

4777 

483o 

39 

20 

9.7i3386 

3438 

3489 

354i 

3592 

3644 

39 

21 

4883 

4936 

4989 

5o42 

5o95 

5i48 

38 

21 

3696 

3747 

3799 

385o 

39O2 

3954 

38 

22 

6201 

5254 

53o7 

536o 

54i2 

5465 

37 

22 

4oo5 

4o57 

4io8 

4i6o 

4211 

4263 

37 

23 

55i8 

557i 

5624 

5677 

573o 

5783 

36 

23 

43i4 

4366 

44i8 

4469 

452i 

4572 

36 

24 

5836 

5888 

594i 

5994 

6047 

6100 

35 

24 

4624 

4675 

4727 

4778 

483o 

488i 

35 

25 

6i53 

6206 

6258 

63n 

6364 

64i7 

34 

25 

4933 

4984 

5o36 

5o87 

5i39 

5i9o 

34 

26 

6470 

6522 

6575 

6628 

6681 

6734 

33 

26 

5242 

5293 

5345 

5396 

5448 

5499 

33 

27 

6787 

6839 

6892 

6945 

6998 

7o5o 

32 

27 

555i 

56o2 

5654 

57o5 

5757 

5  808 

32 

28 

7io3 

7i56 

7209 

7262 

?3i4 

7367 

3i 

28 

586o 

5911 

5962 

6oi4 

6o65 

6n7 

3i 

29 

7420 

7473 

7525 

7578 

763i 

7684 

3o 

29 

6168 

6220 

627I 

6322 

6374 

6425 

3o 

3o 

9.697736 

7789 

7842 

7894 

7947 

8000 

29 

3o 

9.716477 

6528 

6579 

663i 

6682 

6734 

29 

3i 

8o53 

8io5 

8i58 

8211 

8263 

83i6 

28 

3i 

6785 

6836 

6888 

6939 

699i 

7042 

28 

32 

8369 

8421 

8474 

8527 

8579 

8632 

27 

32 

7o93 

7i45 

7i96 

7247 

7299 

735o 

27 

33 

8685 

8737 

8790 

8843 

8895 

8948 

26 

33 

74oi 

7453 

75o4 

7555 

7607 

7658 

26 

34 

9001 

9o53 

9106 

9l59 

9211 

9264 

25 

34 

77°9 

7761 

78l2 

7863 

79i5 

•7966 

25 

35 

93i6 

9369 

9^22 

9474 

9527 

9579 

24 

35 

8oi7 

8069 

8120 

8171 

8223 

8274 

24 

36 

9632 

9685 

9737 

9790 

9842 

9895 

23 

36 

8325 

8376 

8428 

8479 

853o 

858i 

23 

37 

QQ/7 

53 

Iio5 

!i58 

o  T  r\ 

o  o 

Qr, 

8633 

8684 

8735 

8786 

8838 

8889 

22 

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38 
39 

vv^/ 
9.700263 
o578 

o3i5 
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o368 
o683 

0420 
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0788 

•  £  1  U 

io525 
o84i 

£& 

21 
2O 

°7 
38 
39 

8940 
9248 

8991 
9299 

0/v»' 

9o43 

935o 

9094 
9401 

9i45 
9452 

9196 
95o4 

21 
2O 

4o 

9.700893 

0946 

0998 

io5i 

no3 

n55 

I9 

4o 

9.7i9555 

9606 

9657 

9708 

976o 

9811 

1  9 

4i 

1208 

1260 

i3i3 

i365 

1418 

1470 

18 

4i 

9862 

99i3 

9964 

..16 

..67 

.118 

18 

42 

i523 

i575 

1628 

1680 

i733 

1785 

17 

42 

9.72oi69 

O22O 

027I 

0322 

o374 

0425 

*7 

43 

1837 

1890 

1942 

1995 

2047 

2IOO 

16 

43 

o476 

o527 

o578 

0629 

0680 

0732 

16 

44 

2l52 

22O4 

2257 

23o9 

2362 

24i4 

i5 

44 

o783 

o834 

o885 

o936 

P987 

io38 

i5 

45 

2466 

2519 

2571 

2623 

2676 

2728 

i4 

45 

1089 

n4o 

1191 

1243 

I294 

1  345 

i4 

46 

2781 

2833 

2885 

2938 

2990 

3o42 

i3 

46 

i396 

i447 

1498 

1  549 

1600 

i65i 

i3 

4? 

3095 

3i47 

3i99 

3252 

33o4 

3356 

12 

47 

I702 

i753 

1804 

i855 

I9o6 

J957 

12 

48 

3409 

346i 

35i3 

3566 

36i8 

367o 

II 

48 

20O9 

2060 

2III 

2162 

22l3 

2264 

II 

49 

3722 

3775 

3827 

3879 

3932 

3984 

IO 

49 

23i5 

2366 

24l7 

2468 

25i9 

2570 

10 

5o 

9.704036 

4o88 

4i4i 

4i93 

4245 

4297 

9 

5o 

9.  7-22621 

2672 

2723 

2774 

2825 

2876 

9 

5i 

435o 

4402 

4454 

45o6 

4559 

46u 

8 

5i 

2927 

2978 

3O29 

3o8o 

3i3o 

3i8i 

8 

52 

4663 

47i5 

4768 

4820 

4872 

4924 

7 

52 

3232 

3283 

3334 

3385 

3436 

3487 

7 

53 

4976 

5029 

5o8i 

5i33 

5i85 

5237 

6 

53 

3538 

3589 

364o 

3691 

3742 

3793 

6 

54 

5290 

5342 

5394 

5446 

5498 

555i 

5 

54 

3844 

3895 

3945 

3996 

4o47 

4o98 

5 

55 

56o3 

5655 

57o7 

5759 

58n 

5863 

4 

55 

4i49 

4200 

425i 

4302 

4353 

44o3 

4 

56 

59i6 

5968 

6020 

6072 

6124 

6176 

3 

56 

4454 

45o5 

4556 

46o7 

4658 

47o9 

3 

57 

6228 

6280 

6333 

6385 

6437 

6489 

2 

57 

476o 

48io 

486i 

4912 

4963 

5oi4 

2 

58 

654i  6593 

6645 

6697 

6749 

6801 

I 

58 

5o65 

5n5 

5:66 

52I7 

5268 

53i9 

I 

59 

6854|69o6 

6958 

7010 

7062 

7ii4 

0 

59 

537o 

5420 

547i 

5522 

5573 

5624 

O 

60"    j  50" 

40" 

30" 

20" 

10" 

. 

60"     50" 

40"  1  30" 

20"  j  10" 

. 

Co-tangent  of  63  Degrees. 

a 

Co-tangent  of  62  Degrees. 

i 

,  r,  2"  3"  4"  5"  6"  7"  8"  9" 

.  (  I''  2"  3"  4"  5"  6"  7"  8"  9" 

I  5  11  1G  21  2G  32  37  42  47 

1  l)  5  10  15  21  26  31  36  41  46 

LOGARITHMIC    SINES. 


1  d 

Sine  of  28  Degrees. 

*  \      Sine  of  29  Degrees. 

5 

0" 

10" 

20" 

30" 

40" 

50" 

* 

0" 

10" 

20" 

30"  '  40" 

50" 

o 

9.671609 

1649 

1688 

1728 

I768 

i8o7 

59 

0 

9.68557I 

5609 

5647 

5685 

5-723 

576i 

59 

I 

i847 

1886 

1926 

i965 

2005 

2o45 

58 

i 

5799 

5837 

5875 

59i3 

595i 

5989 

58 

2 

2084 

2124 

2i63 

2203 

2242 

2282 

57 

2 

6027 

6o65 

6io3 

6i4i 

6178 

6216 

57 

3 

2321 

236i 

2400  2440 

2479 

25i9 

56 

3 

6254 

6292 

633o 

6368 

64o6 

6444 

56 

4 

2558 

2598 

2637 

2677 

2716 

2756 

55 

4 

6482 

65i9 

6557 

6595 

6633 

667i 

55 

5 

2795 

2835 

2874 

29i4 

2953 

2992 

54 

5 

67o9 

6747 

6785 

6822 

6860 

6898 

54 

6 

3o32 

3o7i 

3m 

3i5o 

3190 

3229 

53 

6 

6936 

6974 

7012 

7o49 

7087 

7I25 

53 

7 

3268 

33o8 

3347 

3387 

3426 

3465 

52 

7 

7i63 

7201 

7238 

7276 

73i4 

7352 

52 

8 

35o5 

3544 

3583 

3623 

3662 

3702 

5i 

8 

7389 

7427 

7465 

75o3 

754i 

7578 

5i 

9 

374i 

378o 

382o 

3859 

3898 

3938 

5o 

9 

7616 

7654 

7692 

7729 

7767 

78o5 

5o 

10 

9.673977 

4oi6 

4o56 

4095 

4i34 

4i73 

49 

10 

9.687843 

7880 

7918 

7956 

7993 

8o3i 

49 

ii 

42l3 

4252 

4291 

433i 

437o 

4409 

48 

ii 

8069 

8106 

8i44 

8182 

8220 

8257 

48 

12 

4448 

4488 

4527 

4566 

46o6 

4645 

47 

12 

8295 

8333 

837o 

84o8 

8446 

8483 

47 

i3 

4684 

4723 

4762 

4802 

484i 

488o 

46 

i3 

852i 

8559 

8596 

8634 

8671 

87o9 

46 

i4 

4919 

4959 

4998 

5o37 

5076 

5n5 

45 

i4 

8747 

8784 

8822 

8860 

8897 

8935 

45 

i5 

5i55 

5i94 

5233 

5272 

53ii 

535o 

44 

i5 

8g72 

9010 

9048 

9o85 

9123 

9i6o 

44 

16 

5390 

5429 

5468 

55o7 

5546 

5585 

43 

16 

9198 

9235 

9273 

93n 

9348 

9386 

43 

17 

5624 

5664 

57o3 

5742 

578i 

5820 

42 

17 

9423 

946i 

9498 

9536 

9573 

9611 

42 

18 

5859 

5898 

5937 

5976 

6016 

6o55 

4i 

18 

9648 

9686 

9723 

9761 

9798 

9836 

4i 

19 

6094 

6i33 

6172,6211 

6289 

4o 

19 

9873 

9911 

9948 

9986 

..23 

..61 

4o 

20 

9.676328 

6367 

64o6  6445 

6484 

6523 

39 

20 

9.600098 

oi36 

0173 

02  1  I 

0248 

0286 

39 

21 

6562 

6601 

664o  6679 

67i8 

6757 

38 

21 

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o36i 

o398 

0435 

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0610 

38 

22 

6796 

6835 

68746913 

6952 

6991 

3? 

22 

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o585 

0622 

0660 

0697 

o735 

37 

23 

7o3o 

7o69 

7108  7147 

7i86 

7225 

36 

23 

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0809 

0847 

0884 

O922 

o959 

36 

24 

7264 

73o3 

7342  7381 

7420 

7459 

35 

24 

0996 

io34 

1071 

1108 

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n83 

35 

25 

7498 

7536 

7575  7614 

7653 

7692 

34 

25 

I22O 

1258 

1295 

i332 

1370 

i4o7 

34 

26 

7770 

7809  7848 

7886 

7925 

33 

26 

1  444 

1482 

i5i9 

i556 

i594 

i63i 

33 

27 

7964 

8oo3 

80428081 

8120 

8i58 

32 

27 

1668 

I7o6 

1743 

1780 

1817 

i855 

32 

28 

8197 

8236 

8275  83i4 

8353 

839i 

3i 

28 

1892 

I929 

i966 

2004 

2o4l 

2078 

3i 

29 

843o 

8469 

85o8  8547 

8585 

8624 

3o 

29 

2Il5 

2i53 

2190 

2227 

2264 

2302 

3o 

3o 

9.678663 

8702 

874o  8779 

8818 

8857 

29 

3o 

9.692339 

2376 

24i3 

245o 

2488 

2525 

29 

3i 

8895 

8934 

8973  9012 

9o5o 

9089 

28 

3i 

2062 

2599 

2636 

2674 

2711 

2748 

28 

32 

9128 

9167 

9205  9244 

9283 

9321 

27 

32 

2785 

2822 

2859 

2897 

2934 

297I 

27 

33 

9360 

9399 

9438  9476 

95i5 

9554 

26 

33 

3oo8 

3o45 

3o82 

3119 

3i57 

3i94 

26 

34 

9592 

963i 

9670  9708 

9747 

9786 

25 

34 

323i 

3268 

33o5 

3342 

3379 

34i6 

25 

35 

9824 

9863 

9902  9940 

9979 

..17 

24 

35 

3453 

349o 

3528 

3565 

36o2 

3639 

24 

36 

9-68oo56 

0095 

oi33  0172 

0210 

0249 

23 

36 

3676 

37i3 

375o 

3787 

3824 

386i 

23 

37 

0288 

o326 

o365 

o4o3 

0442 

o48i 

22 

37 

3898 

3935 

3972 

4009 

4o46 

4o83 

22 

38 

o5i9 

o558 

o596 

o635 

o673 

0712 

21 

38 

4l20 

4i57 

4194 

423i 

4268 

43o5 

21 

39 

0750 

0789 

0828 

0866 

0905 

o943 

20 

39 

4342 

4379 

44i6 

4453 

449o 

4527 

20 

4o 

9.680982 

1020 

io59 

io97 

n36 

1174 

I9 

4o 

9.694564 

46oi 

4638 

4675 

47-2 

4?49 

I9 

4i 

I2l3 

I25l 

1290 

i328 

i366 

i4o5 

18 

4i 

4786 

4823 

486o 

4897 

4934 

4971 

18 

42 

i443 

1482 

l52O 

i559 

l597 

i636 

I7 

42 

5007 

5o44 

5o8i 

5n8 

5i55 

5192 

17 

43 

i674 

I7i3 

i75i 

i789 

1828 

1866 

16 

43 

5229 

5266 

53o3 

5339 

5376 

54i3 

16 

44 

1905 

i943 

1981 

2O20 

2o58 

2097 

i5 

44 

545o 

5487 

5524 

556i 

5598 

5634 

i5 

45 

2i35 

2173 

2212 

225o 

2288 

2327 

i4 

45 

567i 

57o8 

5745 

5782 

58i9 

5855 

i4 

46 

2365 

24o3 

2442 

2480 

25i9 

2557 

i3 

46 

5892 

5929 

5966 

6oo3 

6o39 

6o76 

i3 

47 

2595 

2633 

2672 

27IO 

2748 

2787 

12 

47 

6n3 

6i5o 

6187 

6223 

6260 

6297 

12 

48 

2825 

2863 

2902  2940 

2978 

3oi6 

II 

48 

6334 

637o 

6407 

6444 

648  1 

65i7 

II 

49 

3o55 

3o93 

3i3i 

3i7o 

3208 

3246 

10 

49 

6554 

659i 

6628 

6664 

6701 

6738 

IO 

5o 

9.683284 

3323 

336i 

3399 

3437 

3475 

9 

5o 

9.696775 

6811 

6848 

6885 

6921 

6958 

9 

5i 

35i4 

3552 

359o 

3628 

3667 

37o5 

8 

5i 

6995 

7o3i 

7068 

7io5 

7i4i 

7i78 

8 

52 

3743 

378i 

38i9 

3858 

3896 

3934 

7 

52 

725l 

7288 

7325 

736i 

7398 

7 

53 

3972 

4oio 

4o48 

4o87 

4i25 

4i63 

6 

53 

7435 

747i 

75o8 

7545 

758i 

76i8 

6 

54 

4201 

4239 

4277 

43i5 

4353 

4392 

5 

54 

7654 

769i 

7728 

7764 

78oi 

7838 

5 

55 

443o 

4468 

45o6 

4544 

4582 

4620 

4 

55 

7874 

79n 

7947 

7984 

8020 

8007 

4 

56 

4658 

4696 

4735 

4773 

48n 

4849 

3 

56 

8o94 

8i3o 

8167 

8203 

8240 

8276 

3 

57 

4887 

4925 

4963 

5ooi 

5o39 

5o77 

2 

57 

83i3 

8349 

8386 

8423 

8459 

8496 

2 

58 

5n5 

5i53 

5i9i 

5229 

5267 

53o5 

I 

58 

8532 

8569 

86o5 

8642 

8678 

87i5 

I 

59 

5343 

538i 

54i9 

5457 

54o5 

5533 

0 

59 

875i 

8788 

8824 

8861 

8897 

8934 

O 

60" 

50" 

40" 

30"  |  20" 

10" 

a 

60" 

50" 

40" 

30" 

20" 

10" 

^ 

Co-sine  of  61  Degrees. 

Co-sine  of  60  Degrees. 

1 

p  „  .€  1"  2"  3"  4"  5"  6"  7"  8"  9" 

C  1"  2"  3"  4"  5"  6"  7"  8"  9" 

r.rartj  4  8  12  16  19  23  27  31  35 

P.  Part  J  4   7  n  15  19  20  26  30  33 

LOGARITHMIC    TANGENTS. 


53 


1 

Tangent  of  28  Degrees. 

.S 

Tangent  of  29  Degrees. 

§ 

0" 

10" 

20" 

30" 

40" 

50" 

S 

0" 

10" 

20" 

30" 

40" 

50" 

o 

9.725674 

5?25 

5776 

5827 

5878 

5928 

59 

o 

9.743752 

38o2 

385i 

3901 

395l 

4ooo 

59 

5979 

6o3c 

6081 

6i3i 

6182 

6233 

58 

] 

4o5o 

4099 

4i49 

4l99 

4248 

4298 

58 

2 

6284 

6334 

6385 

6436 

6487 

6537 

57 

t 

4348 

4397 

4447 

4496 

4546 

4596 

57 

j 

6588 

663g 

6690 

674o 

679i 

6842 

56 

' 

4645 

4695 

4744 

4794 

4844 

4893 

56 

i 

6892 

6943 

6994 

7o45 

7°95 

7i46 

55 

L 

4943 

4992 

5o42 

5092 

5i4i 

5191 

55 

\ 

7197 

7247 

7298 

7349 

7399 

745o 

54 

f 

5240 

5290 

5339 

5389 

5439 

5488 

54 

6 

75oi 

755i 

76o2 

7653 

77°3 

7754 

53 

6 

5538 

5587 

5637 

5686 

5736 

5785 

53 

7 

T^oS 

7855 

79o6 

7957 

8oo7 

8o58 

52 

7 

5835 

5884 

5934 

5983 

6o33 

6082 

52 

8 

8109 

8i59 

8210 

8261 

83n 

8362 

5i 

8 

6i32 

6182 

623i 

6281 

633o 

638o 

5i 

9 

8412 

8463 

85i4 

8564 

86i5 

8665 

5o 

9 

6429 

6479 

6528 

6577 

6627 

6676 

5o 

10 

9.728716 

8767 

8817 

8868 

8918 

8969 

49 

10 

9.746726 

6775 

6825 

6874 

6924 

6973 

49 

li 

9020 

9070 

9121 

9171 

9222 

9272 

4o 

ii 

7023 

7072 

7122 

7171 

722I 

7270 

48 

12 

9323 

9374 

9424 

9475 

9525 

9576 

47 

12 

73i9 

7369 

74i8 

7468 

75i7 

7567 

47 

1C 

9626 

9677 

9727 

9778 

9828 

9879 

46 

i3 

7616 

7665 

77i5 

7764 

78i4 

7863 

46 

i4 

9929 

9980 

..3o 

..81 

.132 

.182 

45 

i4 

79i3 

7962 

Son 

8061 

8110 

8160 

45 

i5 

9.730233 

0283 

o333 

o384 

o434 

o485 

44 

!5 

8209 

8258 

83o8 

8357 

84o6 

8456 

44 

16 

o535 

o586 

o636 

o687 

o737 

o788 

43 

16 

85o5 

8555 

86o4 

8653 

87o3 

8752 

43 

17 

o838 

0889 

0939 

0990 

io4o 

io9i 

42 

!7 

8801 

885i 

8900 

8949 

8999 

9o48 

42 

18 

n4i 

1191 

1242 

1292 

!343 

i393 

4i 

18 

9o97 

9147 

9196 

9245 

9295 

9344 

4i 

19 

i444 

i494 

1  544 

i595 

i645 

1696 

4o 

I9 

9393 

9443 

9492 

954i 

959i 

964o 

4o 

20 

9-731746 

1796 

1847 

1897 

i948 

1998 

39 

20 

9.749689 

9739 

9788 

9837 

9886 

9936 

39 

21 

2048 

2099 

2149 

22OO 

225o 

2300 

38 

21 

9985 

..34 

..84 

.i33 

.182 

.231 

38 

22 

235i 

2401 

245i 

25O2 

2552 

2602 

37 

22 

9.75o28i 

o33o 

o379 

0428 

o478 

o527 

37 

23 

2653 

2703 

2753 

28o4 

2854 

2904 

36 

23 

o576 

0625 

o675 

0724 

o773 

0822 

36 

24 

2955 

3oo5 

3o55 

3io6 

3i56 

3206 

35 

24 

0872 

O92I 

o97o 

1019 

1069 

1118 

35 

25 

3257 

33o7 

3357 

34o8 

3458 

35o8 

34 

25 

1167 

1216 

1265 

i3i5 

1  364 

i4i3 

34 

26 

3558 

3609 

3659 

37o9 

376o 

38io 

33 

26 

1462 

i5n 

i56i 

1610 

1669 

I7o8 

33 

27 

386o 

3910 

3961 

4on 

4o6i 

4m 

32 

27 

1757 

1806 

i856 

I9o5 

i954 

2003 

32 

28 

4162 

4212 

4262 

43l2 

4363 

44i3 

3i 

28 

2O52 

2IOI 

2l5l 

22OO 

2249 

2298 

3i 

29 

4463 

45i3 

4564 

46r4 

4664 

47i4 

3o 

29 

2347 

2396 

2446 

2495 

2544 

2593 

3o 

3o 

9.734764 

48i5 

4865 

49i5 

4965 

5oi5 

29 

3o 

9.  752642 

2691 

2740 

2789 

2839 

2888 

29 

3i 

5o66 

5n6 

5i66 

52i6 

5266 

53i7 

28 

3i 

2937 

2986 

3o35 

3o84 

3i33 

3i82 

28 

32 

5367 

54i7 

5467 

55i7 

5567 

56!8 

27 

32 

323i 

3280 

333o 

3379 

3428 

3477 

27 

33 

5668 

5718 

5768 

58i8 

5868 

5918 

26 

33 

3526 

3575 

3624 

3673 

3722 

377i 

26 

34 

5969 

6019 

6069 

6n9 

6i69 

6219 

25 

34 

3820 

3869 

39i8 

3967 

4oi6 

4o66 

25 

35 

6269 

6319 

6370 

6420 

647o 

6520 

24 

35 

4n5 

4i64 

4si  1  3 

4262 

43n 

436o 

24 

36 

6570 

6620 

6670 

6720 

677o 

6820 

23 

36 

4409 

4458 

45o7 

4556 

46o5 

4654 

23 

3? 

»  6870 

6921 

6971 

7O2I 

7071 

7I2I 

22 

37 

47o3 

4752 

48oi 

485o 

4899 

4948 

22 

38 

7171 

7221 

72-71 

732I 

737i 

742I 

21 

38 

4997 

5o46 

5o95 

5i44 

5i93 

5242 

21 

39 

7471 

752I 

757i 

762I 

7671 

772I 

2O 

39 

5291 

534o 

5389 

5438 

5487 

5536 

2O 

4o 

9.787771 

782I 

7871 

792I 

797i 

8021 

I9 

4o 

9.755585 

5634 

5682 

573i 

578o 

582Q 

19 

4i 

8071 

8121 

8171 

8221 

827r 

832i 

18 

4i 

5878 

5927 

5976 

6o25 

6o74 

6i23 

18 

42 

837i 

8421 

847i 

852i 

857i 

8621 

J7 

42 

6172 

6221 

62-70 

6319 

6368 

64i6 

*7 

43 

8671 

872I 

877i 

8821 

8871 

8921 

16 

43 

6465 

65i4 

6563 

6612 

6661 

67io 

16 

44 

8971 

9021 

9071 

9121 

9i7i 

9221 

i5 

44 

6759 

6808 

6857 

6905 

6954 

7oo3 

i5 

45 

9271 

932I 

937i 

9420 

947o 

9520 

i4 

45 

7052 

7IOI 

7i5o 

7i99 

7247 

7296 

i4 

46 

.  ',57° 

9620 

967o 

972O 

977o 

9820 

3 

46 

7345 

7394 

7443 

74g2 

754i 

7589 

i3 

47 

9870 

9920 

9969 

.19 

.69 

119 

2 

47 

7638 

7687 

7736 

7785 

7834 

7882 

12 

48 

9.740169 

0219 

0269 

o3i9 

o368 

0418 

I 

48 

793i 

7980 

8029 

8o78 

8127 

8i75 

II 

49 

o468 

o5i8 

o568 

0618 

0668 

07I7 

0 

49 

8224 

8273 

8322 

837i 

84i9 

8468 

IO 

5o 

9.740767 

o8i7 

o867 

o9i7 

096-7 

016 

9 

5o 

9.7585i7 

8566 

86i5 

8663 

8712 

876i 

9 

5i 

1066 

116 

1166 

1216 

265 

3i5 

8 

5i 

8810 

8858 

8907 

8956 

9005 

9o53 

8 

62 

i365 

4i5 

i465 

5i4 

564 

6i4 

7 

52 

9102 

9i5i 

9200 

9248 

9297 

9346 

7 

53 

i664 

714 

i763 

8i3 

863 

9i3 

6 

53 

9395 

9443 

9492 

954i 

959o 

9638 

6 

54 

1962 

2OI2 

2062 

2112 

2161 

2211 

5 

54 

9687 

9736 

9785 

9833 

9882 

9931 

5 

55 

2261 

23n 

236o 

2410 

2460 

25lO 

4 

55 

99-79 

.28 

••77 

.  126 

•  I74 

.223 

4 

56 

2559 

609 

2659 

2709 

2758 

2808 

3 

56 

9.76o272 

0320 

o369 

0418 

o466 

o5i5 

3 

57 

2858 

907 

2957 

ioo7 

3o56 

106 

2 

57 

o564 

0612 

0661 

O7IO 

o758 

o8o7 

2 

58 

3t56 

206 

3255 

33o5 

3355 

34o4 

I 

58 

o856 

0904 

o953 

IOO2 

io5o 

1099 

I 

,  *9 

3454 

5o4 

3553 

36o3 

3653 

702 

O 

59 

11*49 

1196 

1245 

1293 

1  342 

1391 

0 

60" 

ay 

40" 

30" 

20" 

10" 

60" 

50" 

40" 

30" 

20" 

10" 

. 

. 

Co-  tangent  of  6  1  Degrees. 

S 

Co-tangent  of  60  Degrees. 

1 

p  p  .<  1"  2"  3"  4"  5"  6"  7"  8"  9" 

,<1"  2"  3"  4"  5"  6"  7"  8"  9" 

irl$  5  10  15  20  25  30  35  40  45 

I  5   10  15  20  25  29  34  39  44 

LOGARITHMIC    SIXES. 


J 

Sine  of  30  Degrees. 

.5 

Sine  of  31  Degrees. 

s 

0" 

10" 

20"  |  30"   40" 

50" 

s 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.69897o 

9oo6 

9043^079  9116 

9l52 

69 

09.711839 

1874 

I9o9 

i944 

1979 

201^ 

59 

I 

9i89 

9225 

9262  9298  933^ 

937i 

58 

I 

2o5o 

2o85 

2I2O 

2i55 

2190 

2225 

58 

2 

94o7 

9444 

948o  95i7 

9553 

9589 

57 

2 

2260 

2295 

2330 

2365 

2400 

2434 

57 

2 

9626 

9662 

9699  9735 

9771 

98o8 

56 

*3 

2469 

25o4 

2539 

2574 

2609 

2644 

56 

4 

9844 

988o 

9917 

9953 

999° 

.  ,26 

55 

4 

2679 

2714 

2749 

2784 

28i9 

2854 

55 

e 

9  .  700062 

°°99 

oi35  0171 

0208 

0244 

54 

f. 

2889 

2924 

2959 

2994 

3o29 

3o63 

54 

6 

0280 

o3i7 

o353o389 

042  5 

0462 

53 

6 

3o98 

3i33 

3i68 

32o3 

3238 

3273 

53 

7 

o498 

o534 

o57i  0607 

o643 

0680 

52 

7 

33o8 

3343 

3377 

34i2 

3447 

3482 

52 

8 

0716 

0752 

o788  0825 

0861 

0897 

5i 

8 

35i7 

3552 

3587 

362i 

3656 

369i 

5i 

9 

o933 

o97o 

1006  1042 

1078 

in5 

5o 

9 

3726 

376i 

3796 

383o 

3865 

39oo 

5o 

10 

9.701151 

n87 

I22S 

1259 

I296 

i332 

49 

10 

9.7i3935 

397o 

4oo5 

4o39 

4074 

4109 

49 

ii 

1  368 

i4o4 

i44o 

1477 

i5i3 

1  549 

48 

ii 

4i79 

4213 

4248 

4283 

43i8 

48 

12 

i585 

1621 

i658 

i694 

1730 

1766 

47 

12 

4352 

4387 

4422 

4457 

449i 

4526 

47 

i3 

1802 

i838 

i874 

i9n 

i947 

i983 

46 

i3 

456i 

4596 

463o 

4665 

4700 

4735 

46 

i4 

2019 

2o55 

2091 

2127 

2164 

22OO 

45 

i4 

4769 

4804 

4839 

4873 

4908 

4943 

45 

i5 

2236 

2272  23<>8 

2344 

238o 

2416 

44 

i5 

4978 

5OI2 

5o47 

5o82 

5n6 

5i5i 

44 

16 

2452 

2488  2524 

256i 

2597 

2633 

43 

16 

5  1  86 

5220 

5255 

529o 

5324 

5359 

43 

17 

2669 

27o5 

274l 

2777 

28i3 

2849 

42 

17 

5394 

5428 

5463 

5498 

5532 

5567 

42 

18 

2885 

292I 

2957 

2993 

3o29 

3o65 

4i 

i  .8 

56o2 

5636 

567i 

57o5 

574o 

5775 

4i 

i9 

3ioi 

3i37  3173 

3209 

3245 

3281 

4o 

I9 

5809 

5844 

5878 

59i3 

5948 

5982 

4o 

20 

9.703317 

3353  3389 

3425 

346i 

3497 

39 

20 

9.716017 

6o5i 

6086 

6121 

6i55 

6i9o 

39 

21 

3533 

3569  36o5 

364i 

3677 

37i3 

38 

21 

6224 

6259 

6293 

6328 

6362 

6397 

38 

22 

3749 

3784  3820 

3856 

3892 

3928 

37 

22 

6432 

6466 

65oi 

6535 

6  5  70 

66o4 

37 

23 

3964 

4ooo4o36 

4o72 

4  1  08 

4i44 

36 

23 

6639 

6673 

6708 

6742 

6777 

6811 

36 

24 

4179 

42i5  425i 

4287 

4323 

4359 

35 

24 

6846 

6880 

69i5 

6949 

6984 

7oi8 

35 

25 

4395 

443i  4466 

45o2 

4538 

4574 

34 

25 

7o53 

7087 

7122 

7i56 

7191 

7225 

34 

26 

46  1  o 

4646  4682 

4717 

4753 

4789 

33 

26 

7259 

7294 

7328 

7363 

7397 

7432 

33 

27 

4825 

48614896 

4932 

4968 

5oo4 

32 

27 

7466 

75oo 

7535 

7669 

76o4 

7638 

32 

28 

5o4o 

5075  5m 

5i47 

5i83 

5219 

3i 

28 

76*8 

7707 

7  74  1 

7776 

78io 

7844 

3i 

29 

5254 

529o5326 

5362 

5397 

5433 

3o 

29 

7879 

79i3 

7948 

7082 

8016 

8o5i 

3o 

3o 

9.705469 

55o5  554o 

5576 

56i2 

5648 

2Q 

3o 

9.71808  5 

8n9 

8i54 

8188 

8223 

8257 

29 

3i 

5683 

57i9i5755 

579o 

5826 

5862 

28 

3i 

8291 

8326 

836o 

8394 

8429 

8463 

28 

32 

5898 

5933  5969 

6oo5 

6o4o 

6076 

27 

32 

8497 

853i 

8566 

8600 

8634 

8669 

27 

33 

6112 

61476183 

62I9 

6254 

6290 

26 

33 

87o3 

8737 

8772 

8806 

884o 

8874 

26 

34 

6326 

636i  6397 

6433 

6468 

65o4 

25 

34 

89o9 

8943 

8977 

9on 

9o46 

9080 

25 

35 

6539 

65766611 

6646 

6682 

6718 

24 

35 

9n4 

9i48 

9i83 

92I7 

92-5l 

9285 

24 

36 

6753 

6789*6824 

6860 

6895 

693i 

23 

36 

9320 

9354 

9388 

9456 

949i 

23 

37 

6967 

7oo2  7038 

7o73 

7I09 

7i45 

22 

37 

9525 

9559 

9593 

9627 

9662 

9696 

22 

38 

7i8o 

7216  7251 

7287 

7322 

7358 

21 

38 

973o 

9764 

9798 

9833 

9867 

99oi 

21 

39 

7393 

7429  7464 

7535 

757i 

20 

39 

9935 

9969 

...3 

..38 

..72 

.  106 

20 

4o  9.  707606 

7642:7677 

77?3 

7748 

7784 

I9 

4o 

9.  720140 

oi74 

0208 

0242 

0276 

o3n 

I9 

4i    7819 

7855  789o 

7926 

796i 

7997 

18 

4i 

o345 

o379 

o4i3 

o447 

0481 

o5i5 

18 

42;    8o32 

8068,  8  1  o3 

8i39 

8i74 

8210 

17 

42 

o549 

o583 

o6i7 

o652 

0686 

O72O 

17 

43]   8245 

82So83i6 

835i 

8387 

8422 

16 

43 

o754 

o788 

0822 

o856 

0890 

0924 

16 

44 

8458 

8493  8528 

8564 

8599 

8635 

i5 

44 

o958 

0992 

1026 

1060 

1094 

1128 

i5 

45 

867o 

87o5!874i 

8776 

8811 

8847 

i4 

45 

1162 

1196 

1230 

1264 

1298 

i332 

i4 

46 

8882 

89i88953 

8988 

9024 

9059 

i3 

46 

i366 

i4oo 

i434 

1  468 

1502 

i536 

i3 

47 

9o94 

9i3o  9i65 

92OO 

9236 

9271 

12 

4? 

1570 

1604 

i638 

1672 

1706 

1740 

12 

48 

93o6 

93429377 

94l2 

9448 

9483 

II 

48 

1774 

1808 

1842 

1876 

1910 

i944 

II 

49 

95i8 

95539589 

9624 

9659 

9695 

10 

49 

1978 

2OI2 

2046 

2080 

2Il4 

2148 

10 

5o 

9.7o973o 

9765 

9800 

9836 

9871 

9906 

9 

5o 

9.  722181 

22l5 

2249 

2283 

23l7 

235i 

9 

5i 

9941 

9977 

..12 

..47 

..82 

.118 

8 

5i 

2385 

24l9 

2453 

2487 

2520 

2554 

y 

52 

9.7ioi53 

0188 

0223 

O259 

0294 

0329 

7 

52 

2588 

2622 

2656 

2690 

2724 

2757 

7 

53 

o364 

0399 

o435 

0470 

o5o5 

o54o 

6 

53 

2791 

2825 

2859 

2893 

2927 

296o 

6 

54 

o575 

0611 

o646 

0681 

0716 

0751 

5 

54 

2994 

3028 

3o62 

3096 

3i3o 

3i63 

5 

55 

o786 

0822 

o857 

o892 

O927  0962 

4 

55 

3197 

323! 

3265 

3299 

3332 

3366 

4 

56 

°997 

1032 

io67 

no3 

n38  1173 

3 

56 

3400 

3434 

3468 

35oi 

35353569 

3 

57 

1208 

1243 

1278 

i3i3 

1  348  1  383 

2 

57 

36o33636367o 

370437383771 

2 

58 
59 

i4i9 
i629 

1454 

1  664 

i489 
1699 

ID24 

1734 

i559  i594 
i769  1804 

O 

58 
59 

38o5  3839  3873  39o6  394o  3974 
%  4007  4o4i  4075  4io9'4i42  4176 

I 

0 

60" 

50" 

40"   30" 

20"   10" 

S3 

60"     50"   40"   30"   20"  i  10"  j  „. 

Co-sine  of  59  Degrees. 

1 

Co-sine  of  58  Degrees.     & 

T,  Pflrt$  1"  2"  3"  4"  5"  6"  7"  8"  9" 
)  4   7  11  14  18  21  25  29  32 

P  PartJ  l"  ~"  3"  4"  5"  6"  7"  8"  9" 

LOGARITHMIC    TANGENTS. 


e 

Talent  of  30  Degrees. 

d 

Tangent  of  31  Degrees. 

s 

0"    |  10" 

20"  |  30"   40"   50" 

2 

0" 

10" 

20" 

30" 

40"   50" 

o 

0.761439 

i488 

l537 

i585 

i634  1682 

59 

o 

9.778774 

8821 

8869 

89i7 

8964(9012 

59 

I 
2 

i73i 

2023 

1780 
2071 

1828 
2120 

1877  i925  i974 
2168  2217  2266 

58 
57 

i 

2 

9060 
9346 

9io8 
9394 

9i55 
q44i 

9203 

q489 

925: 
9537 

9298 
9584 

58 

3 

23i4 

2363 

2411 

2460  25o8 

2557 

56 

3 

9632 

9679 

9727 

9775 

9822 

987o 

56 

4 

2606 

2654 

2703 

2751  2800 

2848 

55 

4 

9918 

9965 

..61 

.I08 

.i56 

55 

5 

2897 

2945 

2994 

3o43 

3o9i 

3i4o 

54 

5 

9.780203 

O25l 

0299 

o346 

o394 

o44i 

54 

6 

3i88 

3237 

3285 

3334 

3382 

343  1 

53 

6 

0489 

o537 

o584 

o632 

o679 

0727 

53 

7 

3479 

3528 

3576 

3625 

3673 

3722 

52 

7 

o775 

0822 

o87o 

°9J7 

o965 

ioi3 

52 

8 

377o 

3819 

3867 

39i6 

3964 

4oi3 

5i 

8 

1060 

1108 

ii55 

1203 

1250 

I298 

5i 

9 

4o6i 

4no 

4i58 

4207 

4255 

43o4 

5o 

9 

1  346 

i393 

i44i 

i486 

i536 

i583 

5o 

IO 

9  ,764352 

44oo 

4449 

4497 

4546 

4594 

49 

10 

9.781631 

1678 

I726 

i774 

1821 

i869 

49 

ii 

4643 

469i 

•4-v4o 

4788 

4836 

4885 

48 

ii 

1916 

1964 

2OII 

2059 

2106 

2i54 

12 

4933 

4982 

5o3o 

5o79 

5l27 

5I?5 

47 

12 

22OI 

2249 

2296 

2344 

239i 

2439 

47 

i3 

5224 

5272 

532i 

5369 

54i8 

5466 

46 

i3 

2486 

2534 

258i 

2629 

2676 

2724 

46 

i4 

55i4 

5563 

56ii 

566o 

57o8 

5756 

45 

i4 

2771 

2819 

2866 

2914 

2961 

Soog 

45 

i5 

58o5 

5853 

59oi 

595o 

5998 

6047 

44 

i5 

3o56 

3io4 

3i5i 

3i99 

3246 

3294 

44 

16 

6095 

6i43 

6i92 

6240 

6288 

6337 

43 

16 

334i 

3388 

3436 

3483 

353i 

3578 

43 

17 

6385 

6433 

6482 

653o 

6578 

6627 

42 

iy 

3626 

3673 

372I 

3768 

38i6 

3863 

42 

18 

6675 

6723 

6772 

6820 

6868 

69i7 

4i 

18 

39io 

3958 

4oo5 

4o53 

4ioo 

4i48 

4i 

19 

6965 

7oi3 

7062 

7110 

7i58 

7207 

4o 

i9 

4i95 

4242 

4290 

4337 

4385 

4432 

4o 

20 

9,767255 

73o3 

7352 

74oo 

7448 

7496 

39 

20 

9.784479 

4527 

4574 

4622 

4669 

4716 

39 

21 

7545 

7593 

7641 

769° 

7738 

7786 

38 

21 

4764 

48ii 

4859 

4906 

4953 

5ooi 

38 

22 

7834 

7883 

793i 

7979 

8o27 

8076 

37 

22 

5o48 

SogS 

5i43 

5190 

5238 

5285 

37 

23 

8124 

8172 

8221 

8269 

83i7. 

8365 

36 

23 

5332 

538o 

5427 

5474 

5522 

5569 

36 

24 

84i4 

8462 

85io 

8558 

8606 

8655 

35 

24 

56i6 

5664 

57ii 

5758 

58o6 

5853 

35 

25 

87o3 

875i 

8799 

8848 

8896 

8944 

34 

25 

5900 

5948 

5995 

6042 

6o9o 

6137 

34 

26 

8992 

9o4o 

9o89 

9i37 

9i85 

9233 

33 

26 

6i84 

6232 

6279 

6326 

6374 

6421 

33 

27 

9281 

933o 

9378 

9426 

9474 

9522 

32 

27 

6468 

65i6 

6563 

6610 

6657 

6705 

32 

28 

957i 

96i9 

9667 

97i5 

9763 

98ll 

3i 

28 

6752 

6799 

6847 

6894 

694i 

6988 

3i 

29 

9860 

9908 

9956 

...4 

..52 

.  IOO 

3o 

29 

7o36 

7083 

7i3o 

7178 

7225 

7272 

3o 

3o 

9  77oi48 

0197 

0245 

0293 

o34i 

0389 

29 

3o 

9.787319 

7367 

74i4 

746i 

75o8 

7556 

29 

3i 

o437 

o485 

o534 

o582 

o63o 

0678 

28 

3i 

7603 

765o 

7697 

7745 

7792 

7839 

28 

32 

0726 

0774 

0822 

0870 

o9i9 

0967 

27 

32 

7886 

7934 

798i 

8028 

8o75 

8122 

27 

33 

ioi5 

io63 

IIII 

n59 

I2O7 

1255 

26 

33 

8170 

8217 

8264 

83n 

8359 

84o6 

26 

34 

i3o3 

i35i 

i399 

1  448 

i496 

1  544 

25 

34 

8453 

85oo 

8547 

8595 

8642 

8689 

25 

35 

i5g2 

i64o 

1688 

i736 

i784 

i832 

24 

35 

8736 

8783 

883o 

8878 

8925 

80.72 

24 

36 

1880 

1928 

i976 

2024 

2072 

2120 

23 

36 

9oi9 

9o66 

9114 

9161 

92O8 

9255 

23 

37 

2168 

2216 

2264 

23l2 

236i 

2409 

22 

37 

93o2 

9349 

9397 

9444 

949i 

9538 

22 

38 

2457 

25o5 

2553 

2601 

2649 

2697 

21 

38 

9585 

9632 

9679 

9727 

9774 

9821 

21 

39 

2745 

2793 

2841 

2889 

2937 

2985 

2O 

39 

9868 

99i5 

9962 

...9 

..57 

.  io4 

2O 

4o 

9,773o33 

3o8i 

3i29 

3i77 

3225 

3273 

I9 

4o 

9.79oi5i 

oi98 

O245 

0292 

o339 

o386 

19 

4i 

332i 

3369 

34i7 

3465 

35i2 

356o 

18 

4i 

o434 

o48i 

o528 

o575 

0622 

o669 

18 

42 

36o8 

3656 

3704 

3752 

38oo 

3848 

17 

42 

0716 

o763 

0810 

o857 

o9o5 

0952 

17 

43 

3896 

3944 

3992 

4o4o 

4o88 

4i36 

16 

43 

°999 

io46 

io93 

n4o 

1187 

1234 

16 

44 

4i84 

4232 

4280 

4328 

4375 

4423 

i5 

44 

1281 

i328 

i375 

1422 

i469 

i5i6 

i5 

45 

447i 

45i9 

4567 

46i5 

4663 

4711 

i4 

45 

i563 

1611 

i658 

1705 

1752 

i799 

i4 

46 

4759 

4807 

4855 

4902 

495o 

4998 

i3 

46 

1  846 

1893 

1940 

1987 

2034 

2081 

i3 

47 

5o46 

5o94 

5i42 

5i9o 

5238 

5286 

12 

47 

2128 

2175 

2222 

2269 

23i6 

2363 

12 

48 

5333 

538i 

5429 

5477 

5525 

5573 

II 

48 

2410 

2457 

25o4 

255i 

2598 

2645 

II 

49 

562i 

5668 

57i6 

5764 

58i2 

586o 

IO 

49 

260,2 

2739 

2-786 

2833 

2880 

20.27 

IO 

5o 

9.7759o8 

5956 

6oo3 

6o5i 

6o99 

6i47 

9 

5o 

9-792974 

3021 

3o68 

3ii5 

3i62 

3209 

9 

5i 

6i95 

6243 

629o 

6338 

6386 

6434 

8 

5i 

3256 

33o3 

335o 

3397 

3444 

3491 

8 

52 

6482 

6529 

6577 

6625 

6673 

6721 

7 

52 

3538 

3585 

3632 

3679 

3726  3773 

7 

53 

6768 

6816 

6864 

69I2 

696o 

7007 

6 

53 

38i9 

3866 

39i3 

3960  4007  4o54 

6 

54 

7o55 

7io3 

7i5i 

7i99 

7246 

7294 

5 

54 

4ioi 

4i48  4195 

42424289^336 

5 

55 

7342 

739o 

7437 

7485 

7533 

758i 

4 

55 

4383 

443o  4476  4523  4570  4617 

4 

50 

7628 

7676 

7724 

7772 

7819 

7867 

3 

56 

4664 

47ii  4758  48o5.4852  48oo 

3 

57 

58 

79i5 

8201 

796318010 
82498297 

8o588io6 
8344  8392 

8i54 
844o 

2 

I 

57 

58 

4946 
5227 

4992  5o39 
5274532i 

5o865i335i8o 
536754i4546i 

2 

I 

59 

8488 

85358583 

863i  86788726 

O 

59 

55o8 

5555  56o2  5649  56965742 

O 

W 

50"   40" 

30"   20"   10" 

a 

60"     50"   40"   30"   20"   10" 

rt 

Co-tangent  of  59  Degrees. 

•fl 

Co-tangent  of  58  Degrees. 

1 

pp.  f  U"  2"  3"  4"  5"  6"  7"  8"  9" 
t11}  5  10  14  19  24  29  34  39  43 

PPavtU"  2"  3"  4"  5"  6"  7"  8"  9" 
m\  5  9  14  19  24  28  33  38  43 

LPGARITHMIC    SINES. 


.s 

Sine  of  32  Degrees. 

d 

•*-* 

Sine  of  33  Degrees 

s 

a 

10" 

20" 

30" 

40" 

50" 

is 

0" 

10" 

20" 

30" 

40'' 

50" 

0 

9.724210 

4243 

4277 

43u 

4344 

4378 

59 

0 

9.  736io9 

6i4i 

6i74 

6206 

6238 

)27I 

59 

il   44is 

4445 

4479  45i3 

4546 

458o 

58 

i 

63o3 

6336 

6368 

64oo 

6433 

6465 

58 

2 

46i4 

4647 

468  1 

47i5 

4748 

4782 

57 

2 

(•498 

653o 

6562 

6595 

6627 

665g 

57 

3 

48i6 

4849 

4883; 

4917 

495o 

4984 

56 

3 

6692 

6724 

6757 

6789 

6821 

6854 

56 

4    6017 

5o5i 

5o85;5n8 

5:52 

5i85 

55 

4 

6886 

6918 

695! 

6983 

7oi5 

7o48 

55 

5 

5219 

5253 

52865320 

5353 

5387 

54 

5 

7080 

7112 

7i45 

7177 

7209 

7241 

54 

6 

5420 

5454 

54885521 

5555 

5588 

53 

6 

7274 

73o6 

7338 

737i 

74o3 

7435 

53 

7 

5622 

5655 

5689 

5722 

5756 

5789 

52 

7 

7467 

75oo  7532 

7564 

7597 

7629 

52 

8 

5823 

5856 

5890 

5923 

5957 

599o 

5i 

8 

7661 

7693 

7726 

7758 

779° 

7822 

5i 

9 

6024 

6057 

6091 

6124 

6i58 

6191 

5o 

9 

7855 

7887 

7919 

795i 

7983 

8016 

5o 

10 

9  726225 

6258 

6292 

6325 

6359 

6392 

49 

10 

9.788048 

8080 

81  12 

8i45 

8177 

8209 

49 

ii 

6426 

6459 

6493  6526 

656o 

6593 

48 

ii 

8241 

8273 

83o6 

8338 

837o 

8402 

48 

12 

13 

6626 

6827 

6660 
6860 

6693  6727 
6894  6927 

6760 
6961 

6794 
6994 

47 
46 

12 

i3 

8434 
8627 

8466 
8659 

84998531 
8692  8724 

8563 
8756 

8595 

8788 

47 
46 

i4 

7027 

7061 

7094 

7128 

7161 

7194 

45 

i4 

8820 

8852 

8884 

8917 

8949 

898i 

45 

i5 

7228 

7261 

7294 

7328 

736i 

7394 

44 

i5 

9oi3 

9o45 

9077 

9109 

9141 

9I?3 

44 

16 

7428 

746i 

7494 

7528 

756i 

7594 

43 

16 

9206  9238 

9270 

9302 

9334 

9366 

43 

J7 

7628 

7661 

7694 

7728 

7761 

7794 

42 

J7 

9398 

943o 

9462 

9494 

9526 

9558 

42 

18 

7828 

7861 

7894 

7928 

7961 

7994 

4i 

18 

•959o 

9622 

9654 

9687 

97i9 

975i 

4i 

J9 

8027 

8061 

8094  8127 

8161 

8i94 

4o 

J9 

9783 

98i5 

9847 

9879 

99n 

9943 

4o 

20 

9.728227 

8260 

8294 

83s7 

836o 

8393 

39 

20 

9.739975 

...7 

..39 

..71 

.io3 

.i35 

39 

21 

8427 

846o 

8493 

8536 

856o 

8593 

38 

21 

9.740167 

oi99 

023l 

0263 

0295 

o327 

38 

22 

8626 

8659 

8692 

8726 

8759 

8792 

37 

22 

o359 

o39i 

0423 

o455 

o487 

o5i9 

37 

23 

8825 

8858 

8892 

8925 

8958 

899i 

36 

23 

o55o 

o582 

0614 

o646 

0678 

o7io 

36 

24 

9024 

9o58 

9091 

9124 

9i57 

9i9o 

35 

24 

0742 

0774 

0806 

o838 

0870 

0902 

35 

25 

9223 

9257 

9290^323 

9356 

9389 

34 

25 

o934 

o966 

0997 

1029 

1061 

io93 

34 

26 

9422 

9455 

9489^522 

9555 

9588 

33 

26 

1125 

n57 

1189 

1221 

1253 

1284 

33 

27 

9621 

9654 

9687  9720 

9753 

9787 

32 

27 

i3i6 

1  348 

i38o 

l4l2 

1  444 

i476 

32 

28 

0820 

9853 

9886  9919 

9952 

9985 

3i 

28 

i5o8 

!539 

1571 

i6o3 

i635 

i667 

3i 

29  9.780018 

oo5i 

0084  0117 

oi5o 

oi83 

3o 

29 

i699 

i73o 

1762 

I794 

1826 

i858 

3o 

3o  9.730217 

0250 

0283 

o3i6 

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o382 

29 

3o 

9.74i889 

I92I 

i953 

1985 

2017 

2049 

29 

3i 

o4i5 

o448 

0481 

o5i4 

o547 

o58o 

28 

3i 

2080 

2112 

2i44 

2176 

2207 

2239 

28 

32 

o6i3 

o646 

0679  0712 

o745 

0778 

27 

32 

2271 

2303 

2335 

2366 

2398 

243o 

27 

33 

0811 

o844 

0877  0910 

o943 

o976 

26 

33 

2462 

2493 

2525 

2557 

2589 

2620 

26 

34 

1009 

1042 

1075 

' 

1108 

n4i 

n73 

25 

34 

2652 

2684 

2715 

2747 

2779 

2811 

25 

35 

1206 

1239 

1272 

i3o5 

i338 

i37i 

24 

35 

2842 

2874 

2906 

2937 

2969 

3ooi 

24 

36 

i4o4 

1437 

1470 

i5o3 

i536 

i569 

23 

36 

3o33 

3o64 

3096 

3i28 

3!59 

3i9i 

23 

37 

1602 

i634 

1667 

1700 

i733 

1766 

22 

37 

3223 

3254 

3286 

33i8 

3349 

338i 

22 

38 

1799 

i832 

i865 

1897 

1930 

i963 

21 

38 

34i3 

3444 

3476 

35o8 

3539 

357i 

21 

39 

1996 

2029 

2062 

2095 

2127 

2160 

20 

39 

36o2 

3634 

3666 

3697 

3729 

376i 

20 

4o 

9.732i93 

2226 

2259 

2292 

2325 

2357 

19 

4o 

9.743792 

3824 

3855 

3887 

39i9 

395o 

19 

4i 

2390 

2423 

2456 

2489 

2521 

2554 

18 

4i 

3982 

4oi3 

4o45 

4077 

4io8 

4i4o 

18 

42 

2587 

262O 

2653 

2685 

27l8 

2751 

17 

42 

4:71 

4203 

4234 

4266 

42-97 

4329 

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43 

2784 

28l6 

2849 

2882 

29l5 

2948 

16 

43 

436i 

4392 

4424 

4455 

4487 

45i8 

16 

44 

2980 

3oi3 

3o46 

3o79 

3iu 

3i44 

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44 

455o 

458x 

46i3 

4644 

4676 

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i5 

45 

3i?7 

3210 

3242 

3275 

3  3  08 

334o 

i4 

45 

4739 

4770 

4802 

4833 

4865 

4896 

i-4 

46 

3373 

3406 

3439 

347i 

35o4 

3537 

i3 

46 

4928 

4959 

4991 

5022 

5o54 

5o85 

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4? 

3569 

36o2 

3635 

3667 

3700 

3733 

12 

47 

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5i48 

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5211 

5243 

5274 

12 

48 

3765 

3798 

383i 

3863 

3896 

3929 

II 

48 

53o6 

5337 

5369 

5400 

543i 

5463 

II 

49 

3961 

3994 

4027 

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4092 

4i25 

10 

49 

5494 

5526 

5557 

5589 

5620 

565i 

10 

5o 

9.734157 

4190 

4222 

4255 

4288 

4320 

9 

5o 

9.  745683 

5714 

5746 

5777 

58o8 

584o 

9 

5i 

4353 

4386 

44i8 

445  1 

4483 

45i6 

8 

5i 

587i 

59o3 

5934 

5965 

5997 

6028 

52 

4549 

458i 

46i4 

4646 

4679 

4711 

7 

52 

6060 

6o9i 

6122 

6i54 

6i85 

6216 

7 

53 

4744 

4777 

4809 

4842 

4874 

49Q7 

6 

53 

6248 

6279 

63io 

6342 

6373 

64o4 

6 

54 

4939 

4972 

5oo4 

5o37 

5069 

5102 

5 

54 

6436 

6467 

6498 

653o 

656i 

6592 

5 

55 

5i35 

5i67 

5260 

5232 

5265 

5297 

4 

55 

6624 

6655 

6686 

6718 

6749 

678o 

4 

56 

533o 

5362 

5395 

5427 

546o 

5492 

3 

56 

6812 

6843 

6874 

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6968 

3 

1 

57 

5525 

5557 

559o 

5622 

5655 

5687 

2 

57 

6999 

7o3i 

7062 

7093 

7I2^ 

7i56  2 

58 

5719 

5752 

5784 

58i7 

5849 

5882 

I 

58 

7187 

7218 

7249 

7281 

73l2 

7343 

~- 

59 

5gi4 

5947 

5979 

6011 

6o44 

6076 

o 

59 

7374 

74o6 

7437 

7468 

7499 

753o 

o 

60" 

50" 

40' 

30" 

20" 

10" 

d 

60" 

50" 

40" 

30"  |  20" 

10" 

g 

Co-sine  of  57 

Degrees. 

s 

Co-sine  of  56  Degrees. 

& 

„  „  ,<  1"  2"  3"  4" 
P.  Part  ^  3  7  10  13 

5"  fi"  7"  8"  9" 
17  20  23  26  30 

C  1"  2"  3"  4"  5"  6"  7"  8"  9" 
P.  Fart  ^  3   6  10  13  '6  19  22  25  29  j 

LOGARITHMIC      TANGENTS. 


1 

Tangent  of  32  Degrees. 

.s 

Tangent  of  33  Degrees. 

& 

0" 

10" 

20" 

30" 

40" 

50" 

ii 

0" 

10"   20" 

30" 

40" 

OU" 

0 

9.795789 

5836 

5883 

593o 

5977 

6023 

r 

o 

9.8i25i7 

2563 

2610 

2656 

27O2 

2748 

59 

I 

6070 

6117 

6i64 

6211 

6258 

63o4 

58 

i 

2794 

2840 

2886 

2932 

2978 

3024 

Co 

•ft 

635i 

6398 

6445 

6492 

6539 

6585 

57 

2 

3o7o 

3n6 

3i63 

3209 

3255 

33oi 

$7 

3 

6632 

6679 

6726 

6773 

6819 

6866 

56 

3 

3347 

3393 

3439 

3485 

353i 

3577 

56 

4 

6913 

6960 

7007 

7o53 

7IOO 

7i47 

55 

4 

3623 

3669 

37i5 

376i 

38o7 

3853 

55 

5 

7194 

7241 

7287 

7334 

738i 

7428 

54 

5 

3899 

3945 

399i 

4o37 

4o83 

4i29 

54 

6 

?474 

752I 

7568 

76i5 

•7662 

77o8 

53 

6 

4176 

4222 

4268 

43i4 

436o 

44o6 

53 

7 

7755 

7802 

7849 

7895 

-7942 

7989 

52 

7 

4452 

4498 

4544 

459o 

4636 

4682 

52 

8 

8o36 

8082 

8129 

8i76 

8223 

8269 

5i 

8 

4728 

4774 

4820 

4866 

49I2 

4958 

5i 

9 

83i6 

8363 

8409 

8456 

85o3 

855o 

5o 

9 

5oo4 

5o5o 

5o96 

5i42 

5i88 

5234 

5o 

10 

9.798596 

8643 

8690 

8737 

8783 

883o 

49 

10 

9.  8i528o 

5326 

537i 

54i7 

5463 

55o9 

49 

ii 

8877 

8923 

89-70 

9OI7 

9063 

9IIO 

48 

ii 

5555 

56oi 

5647 

5693 

5739 

5785 

48 

12 

9l57 

9204 

9260 

9297 

9344 

939o 

47 

12 

583i 

5877 

5923 

5969 

6oi5 

6061 

47 

i3 

9437 

9484 

953o 

9577 

9624 

967o 

46 

i3 

6107 

6i53 

6i99 

6245 

629i 

6336 

46 

i4 

9717 

9764 

9810 

9857 

99°4 

995o 

45 

i4 

6382 

6428 

6474 

652O 

6566 

6612 

45 

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9997 

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44 

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6658 

67o4 

675o 

6796 

6842 

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44 

16 

9.800277 

0324 

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43 

16 

6933 

6979 

7025 

7o7i 

7n7 

7i63 

43 

17 

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o6o3 

o65o 

o697 

o743 

°79° 

42 

17 

7209 

7255 

73oi 

7347 

7392 

7438 

42 

18 

o836 

o883 

ogSo 

o976 

1023 

IO7O 

4i 

18 

7484 

753o 

7576 

7622 

7668 

77i4 

4i 

19 

ii  16 

n63 

1209 

1256 

i3o3 

1  349 

4o 

:9 

7759 

78o5 

785i 

7897 

7943 

7989 

4o 

20 

9.801396 

1442 

1  489 

i535 

i582 

l629 

39 

20 

9.8i8o35 

8081 

8126 

8l72 

82!8 

8264 

39 

21 

i675 

1722 

i768 

i8i5 

1862 

1908 

38 

21 

83io 

8356 

8402 

8447 

8493 

8539 

38 

22 

1955 

20OI 

2048 

2094 

2141 

2l87 

37 

22 

8585 

863i 

8677 

8722 

8768 

8814 

37 

23 

2234 

228l 

2327 

2374 

2420 

2467 

36 

23 

8860 

89o6 

8952 

8997 

9o43 

9o89 

36 

24 

25i3 

256o 

2606 

2653 

2699 

2746 

35 

24 

9i35 

9i8i 

9226 

9272 

93i8 

9364 

35 

25 

%792 

2839 

2886 

2932 

2979 

3o25 

34 

25 

94io 

9455 

95oi 

9547 

9593 

9639 

34 

26 

3072 

3n8 

3i65 

3211 

3258 

33o4 

33 

26 

9684 

973o 

9776 

9822 

9868 

99i3 

33 

27 

335i 

3397 

3444 

3490 

3537 

3583 

32 

27 

9959 

...5 

..5i 

..96 

.142 

.188 

32 

28 

363o 

3676 

3723 

3769 

38i6 

3862 

3i 

28 

9.  820234 

0280 

o325 

o37i 

0417 

o463 

3i 

29 

39o9 

SgSS 

4ooi 

4o48 

4094 

4i4i 

3o 

29 

o5o8 

o554 

0600 

o646 

o69i 

o737 

3o  I 

3c 

9.804187 

4234 

4280 

4327 

4373 

4420 

29 

3o 

9.  820783 

o829 

o874 

0920 

0966 

1012 

29 

3i 

4466 

45i3 

4559 

46o5 

4652 

4698 

28 

3i 

io57 

no3 

n49 

n95 

1240 

1286 

28 

32 

4745 

4791 

4838 

4884 

493o 

4977 

27 

32 

i332 

i377 

1423 

i469 

i5i5 

i56o 

27 

33 

5o23 

5070 

5n6 

5i63 

5209 

5255 

26 

33 

1606 

i652 

i697 

i743 

1789 

i835 

26 

34 

53o2 

5348 

5395 

544i 

5487 

5534 

25 

34 

.  1880 

I926 

I972 

20I7 

ao63 

2I09 

25 

35 

558o 

5627 

5673 

57i9 

5766 

58i2 

24 

35 

2I54 

22OO 

2246 

2292 

•,337 

2383 

Of 

36 

5859 

59o5 

595i 

5998 

6o44 

6o9i 

23 

36;   2429 

2474 

2520 

2566 

261  1 

2657 

23 

37 

6137 

6!83 

623o 

6276 

6322 

6369 

22 

37 

27o3 

2748 

2794 

2840 

2885 

293i 

22 

38 

64i5 

6462 

65o8 

6554 

6601 

6647 

21 

38 

2977 

3022 

3o68 

3n4 

3i59 

32o5 

21 

39 

6693 

674o 

6786 

6832 

6879 

6925 

20 

39 

325i 

3296 

3342 

3387 

3433 

3479 

20 

4o 

9  806971 

7018 

7o64 

7IIO 

7i57 

72o3 

19 

4o 

9.  823524 

357o 

36i6 

366i 

37o7 

3753 

19 

4i 

7249 

7296 

7342 

7388 

7435 

748i 

18 

4i 

3798 

3844 

3889 

3935 

398i 

4026 

18 

42 

7527 

7574 

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7666 

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7759 

17 

42 

4072 

4n7 

4:63 

4209 

4254 

43oo 

J7 

43 

78o5 

785i 

7898 

7944 

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16 

43 

4345 

439i 

4437 

4482 

4528 

4573 

16 

44 

8o83 

8129 

8i76 

8222 

8268 

83i4 

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44 

46i9 

4665 

47io 

4756 

48oi 

4847 

i5 

45 

836i 

8407 

8453 

8499 

8546 

8592 

i4 

45 

4893 

4938 

4984 

5o29 

5o75 

5l20 

i4 

46 

8638 

8685 

873i 

8777 

8823 

887o 

1  3 

46 

5i66 

5212 

5257 

53o3 

5348 

5394 

i3 

4? 

8916 

8962 

9oo8 

9o55 

9101 

9i47 

12 

4? 

5439 

5485 

553i 

5576 

5622 

5667 

12 

48 

9i93 

9240 

9286 

9332 

9378 

9424 

II 

48 

57i3 

5758 

58o4 

5849 

5895 

594o 

II 

49 

9471 

95i7 

9563 

96o9 

9656 

9702 

10 

49 

5986 

6o32 

6o77 

6123 

6168 

62!4 

IO 

5o 

9.  8o9748 

9794 

984o 

9887 

9933 

9979 

9 

5o 

9.826259 

63o5 

635o 

6396 

644  1 

6487 

9 

5i 

9.810025 

0071 

0118 

0164 

O2IO 

0256 

8 

5i 

6532 

6578 

6623 

6669 

6714 

676o 

8 

52 

0302 

0349 

o395 

o44i 

o487 

o533 

7 

52 

68o5 

685i 

6896 

6942 

6087 

7o33 

7 

53 

o58o 

0626 

o672 

0718 

o764 

0810 

6 

53 

7078 

7124 

7i69 

72l5 

7260 

73o6 

6 

54 

0857 

0903 

o949 

o995 

io4i 

1087 

5 

54 

735i 

7397 

7442 

7488 

7533 

7579 

5 

55 

n34 

1180 

1226 

1272 

i3i8 

1  364 

4 

55 

7624 

767o 

77i5 

776i 

7806 

785i 

4 

56 

i4io 

i457 

i5o3 

1  549 

i595 

i64i 

3 

56 

7897 

7942 

7988 

8o33 

8o79 

8124 

3 

57 

1687 

I733 

i78o 

1826 

l872 

I9i8 

2 

57 

8170 

82i5 

8261 

83o6 

835i 

8397 

2 

58 

1964 

2OIO 

2o56 

2IO2 

2i49 

2I95 

I 

58 

8442 

8488 

8533 

8579 

8624 

8669 

1 

59 

224l 

2287 

2333 

2379 

2425 

2471 

O 

59 

87i5 

876o 

8806 

885i 

8897 

8942 

O 

60" 

50"   40" 

30" 

20" 

10" 

d 

60"     50" 

40" 

?0" 

20" 

10" 

Co-tangent  of  57  Degrees. 

i 

Co-tangent  of  56  Degrees. 

1 

pp  ,p"  2"  3"  4"  5"  6"  7"  8"  9"  1 
l™}  5   9  14  19  23  28  33  37  42 

P  Part  5  l"   2//   3"   4//   5"   6"   7"   8"  9" 

trl}  5   9  14  18  23  27  32  37  41 

58 


LOGARITHMIC    SINES 


d 

Sine  of  34  Degrees. 

1 

Sine  of  35  Degrees. 

— 

S 

0" 

10" 

20"   30" 

40" 

50" 

& 

0" 

10" 

20" 

30"  I  40" 

50" 

0 

9.  747662 

7593 

76247666 

7686 

7718 

59 

O 

9.  -768691 

8621 

865l 

868l  8712 

8742 

69 

I 

7749 

7780 

7811  7842 

7874 

79°5 

58 

I 

8772 

8802 

8832 

8862 

8892 

8922 

58 

2 

7936 

7967 

7998  8o3o 

8o6l 

8092 

57 

2 

8962 

8982 

9012 

9042 

9072 

9102 

57 

3 

8i23 

8i54 

8i85  8216 

8248 

8279 

56 

3 

9132 

9162 

9192 

9222 

9262 

9282 

56 

4 

83io 

834i 

8372  84o3 

8434 

8466 

55 

4 

93l2 

9342 

9372 

9402 

9432 

9462 

55 

5 

8497 

8628 

86698690 

8621 

8662 

54 

5 

9492 

9622 

9662 

9682 

9612 

9642 

54 

6 

8683 

87i4 

8745  8777 

8808 

8839 

53 

6 

9-702 

9732 

9762 

9792 

9822 

53 

7 

8870 

8901 

8932  8963  8994 

9026 

52 

7 

9862 

9881 

9911 

9941 

997i 

...  I 

62 

8 
9 

9066 
9243 

9087 
9274 

91189149 
93069336 

9180 
9367 

9212 
9398 

5i 
5o 

8 
9 

9.76oo3i 

O2II 

0061 
0240 

0091 

O270 

OI2I 

o3oo 

oi5i 
o33o 

0181 
o36o 

5i 
5o 

IO 

9-749429 

94609491 

9622 

9553 

9584 

49 

IO 

9.76o39o 

0420 

o45o 

o48o 

0609 

o539 

49 

ii 

96i5 

96469677 

97o8 

9739 

977° 

48 

ii 

o569 

0699 

0629 

0669 

0689 

0718 

48 

12 

98oi 

9832 

9863 

9894 

9926 

9966 

47 

12 

o748 

o778 

0808 

o838 

0868 

0898 

47 

i3 

9987 

..18 

..48 

••79 

.110 

46 

i3 

0927 

o957 

o987 

IOI7 

io47 

1076 

46 

i4 

9.  760172 

0203 

0234 

0266 

0296 

0327 

45 

i4 

1106 

n36 

1166 

1196 

1226 

1266 

45 

i5 

o358 

o389  0420 

o45  1 

0482 

0612 

44 

i5 

1286 

i3i5 

1  345 

i374 

i4o4 

i434 

44 

16 

o543 

o574 

o636 

0667 

0698 

43 

16 

i464 

i4g4 

1623 

i553 

i583 

i6i3 

43 

i7 

o729 

0760  O79I 

0821 

0862 

o883 

42 

17 

1.642 

l672 

I7O2 

I732 

1761 

1791 

42 

18 

o9i4 

0945 

0976 

I007 

1037 

1068 

4i 

18 

1821 

1861 

1880 

1910 

1940 

1969 

4i 

19 

1099 

u3o 

1161 

1192 

1222 

1263 

4o 

i9 

i999 

2029 

2069 

2088 

2118 

2148 

4o 

20 

9.761284 

i3i5 

1  346 

i377 

1407 

i438 

39 

20 

9.762I77 

22O7 

2237 

2267 

2296 

2326 

39 

21 

1469 

1600  i53i 

1661 

1692 

1623 

38 

21 

2356 

2385 

2416 

2445 

2474 

2604 

38 

22 

1  654 

i685 

1716 

i746 

[777 

1808 

37 

22 

2534 

2563 

2693 

2623 

2662 

2682 

37 

23 

1839 

1869 

1900 

1931 

1962 

1992 

36 

23 

27I2 

274l 

277I 

2801 

283o 

2860 

36 

24 

2023 

2064 

2086 

2116 

2146 

2177 

35 

24 

2889 

2919 

2949 

2978 

3oo8 

3o38 

35 

25 

2208 

2238^2269 

2300 

233o 

236i 

34 

25 

3o67 

3o97 

3i26 

3i56 

3i86. 

32i5 

34 

26 

2392 

2423 

2453 

2484 

2616 

2545 

33 

26 

3245 

3274 

33o4 

3333 

3363 

3393 

33 

27 

2676 

2607 

2637 

2668 

2699 

2729 

32 

27 

3422 

3462 

348i 

35n 

354o 

357o 

32 

28 

2760 

2791 

2822 

2862 

2883 

2914 

3i 

28 

36oo 

3629 

3659 

3688 

37i8 

3747 

3i 

29 

2944  2976 

3oo5 

3o36 

3067 

3097 

3o 

29 

3777 

38o6 

3836 

3865 

3895 

3926 

3o 

3o 

5.753i28 

3169 

3i89 

322O 

325i 

3281 

29 

3o 

9.763954 

3984 

4oi3 

4o43 

4072 

4102 

29 

3i 

33i2 

3342 

3373 

34o4 

3434 

3465 

28 

3i 

4i3i 

4i6i 

4190 

4220 

4249 

4279 

28 

32 

3495 

3526 

3557 

3587 

36i8 

3648 

27 

32 

43o8 

4338 

4367 

4396 

4426 

4455 

27 

33 

3679 

3710 

374o 

377i 

38oi 

3832 

26 

33 

4485 

45i4 

4544 

4573 

46o3 

4632 

26 

34 

3862 

3893 

3923 

3954 

3985 

4oi5 

26 

34 

4662 

4691 

4720 

4760 

4779 

48o9 

26 

35 

4o46 

40764107 

4i37 

4i68 

4198 

24 

35 

4838 

4868 

4897 

4926 

4966 

4985 

24 

36    4229 

4269^290 

4320 

435i 

438i 

23 

36 

5oi5 

5o44 

5o74 

5io3 

5i32 

6162 

23 

37    44i2 

44434473 

45o3 

4534 

4564 

22 

37 

5i9i 

6221 

6260 

6279 

53o9 

5338 

22 

38 

4595 

46254656 

4686 

4717 

4747 

21 

38    5367 

5397 

6426 

5456 

5485 

55i4 

21 

39 

4778 

48o8 

4839 

4869 

4900 

493o 

20 

39    5544 

5573 

6602 

5632 

566i 

569o 

20 

4o 

9.7640,60 

4991 

6021 

6062 

6082  5n3 

I9 

4o  9.76572o 

5749 

5778 

6808 

5837 

5866 

I9 

4i 

5i43 

6173 

6204 

5234 

6266 

6296 

18 

4i 

6896 

5925 

5954 

5984 

6oi3 

6042 

1  8 

42 

5326 

5356 

5386 

54i7 

5447 

5478 

i7 

42 

6o72 

6101 

6i3o 

6169 

6189 

6218 

17 

43 

55o8 

5538 

6669 

6699 

6629 

566o 

16 

43 

624-7 

6277 

63o6 

6335 

6364 

6394 

16 

44 

569o 

6721 

575i 

6781 

6812 

5842 

16 

44 

6423 

6462 

648  1 

65n 

654o 

6569 

16 

45 

6872 

59o3 

6933 

5963 

6994 

6024 

i4 

45 

6698 

6628 

6657 

6686 

6716 

6745 

i4 

46 

6o54 

6o85 

6n5 

6i45 

6176 

6206 

i3 

46 

6774 

68o3 

6832 

6862 

6891 

6920 

i3 

47 

6236 

6267 

6297 

6327 

6358 

6388 

12 

47 

6949 

6978 

7oo8 

7o37 

7066 

7o95 

12 

48 

64i8 

6448 

647Q 

6609 

6539 

6670 

II 

48 

7I24 

7i54 

7i83 

7212 

7241 

7270 

II 

49 

6600 

663o666o 

6691 

6721 

6761 

IO 

49 

73oo 

7329 

7358 

7387 

74i6 

7445 

10 

5o 

9.766782 

6812 

6842 

6872 

6903 

6933 

9 

5o 

9-767475 

7604 

7533 

7662 

7691 

7620 

9 

5i 

6963 

6993 

7023 

7o54 

7084 

7114 

8 

Si 

7649 

7679 

77o8 

7737 

7766 

7795 

8 

52 

7  1  44 

7176 

72o5 

7235 

7266 

7296 

7 

62 

7824 

7853 

7882 

79I2 

794i 

797° 

7 

53 

7326 

7356 

7386 

74i6 

7446 

7477 

6 

53 

7999 

8028 

8o57 

8086 

8n5 

8i44 

6 

54 

75o7 

7537 

7567 

7697 

7627 

7668 

5 

54 

8i73 

8203 

8232 

826l 

8290 

83i9 

5 

55 

7688 

7718 

7748 

7778 

7808 

783o 

4 

55 

8348 

8377 

84o6 

8435 

8464 

8493 

4 

56 

7869 

7899 

7Q2Q 

7969 

7989^01^ 

3 

56 

8622 

855i 

858o 

86o9 

8638 

8668 

3 

57 

8o5o 

80808110 

8i4o 

8170  8200 

2 

57 

8697 

8726 

8755 

8784 

88i3 

8842 

2 

5.8    823o 
59|    84n 

8260  820,0 
844i  8471 

832i 
8601 

835i  838i 
853i856i 

I 
O 

58 
59 

887i 
9o45 

8900 
9074 

8929 
9io3 

8968 
9132 

8987 
9161 

9016 
9190 

I 
O 

60" 

50" 

40" 

30" 

20"   10" 

S* 

60" 

50' 

40" 

30" 

20" 

10" 

d 

Co-sine  of  55  Degrees. 

Co-sine  of  54  Degrees. 

p  p  .(  1"  2"  3 

'  4"  5"  6"  7"  8"  9" 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

irl\  3   6   9  12  15  18  21  25  28 

P.  Part  J  3   6   9  12  15  i8  21  24  27 

LOGARITHMIC       ANGENTS. 


1 

Tangent  of  34  Degrees. 

.3 

Tangent  of  35  Degrees. 

0"    |  10" 

20" 

30" 

40" 

50" 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.  82898-7 

9o33 

9078 

912^ 

9l69 

92l5 

59 

o 

9.  845227 

5272 

53i6 

536i 

54o6 

545  1 

59 

I 

9,260 

93o5 

935i 

9396 

944^ 

9487 

58 

i 

5496 

554o 

5585 

563o 

5675 

572O 

58 

2 

9532 

9578 

9623 

9669 

97i4 

9759 

57 

2 

5764 

58o9 

5854 

5899 

5944 

5988 

$7 

^ 

98o5 

985o9895 

c;94l 

9986 

..32 

56 

3 

6o33 

6078 

6i23 

6168 

6212 

6257 

56 

4 

9.83oo77 

OI22 

0168 

02l3 

0258 

o3o4 

55 

4 

63o2 

6347 

639i 

6436 

648! 

652655 

R 

o349 

o395 

o44o 

o485 

o53i 

o576 

54 

5 

657o 

66i5 

6660 

67o5 

675o 

6794 

54 

6 

0621 

o667 

07I2 

o757 

o8o3 

0848 

53 

6 

6839 

6884 

6929 

6973 

7oi8 

7o63 

53 

7 

o893 

o939 

o984 

IO29 

io75 

II2O 

52 

7 

7io8 

7i52 

7197 

-7242 

?287 

733i 

52 

8 

n65 

I2II 

1256 

i3oi 

i347 

l392 

5i 

8 

7376 

7421 

7465 

75io 

7555 

-7600 

5i 

9 

i437 

i483 

i528 

i573 

1619 

1  664 

5o 

9 

7644 

7689 

7734 

7779 

7823 

7868 

5o 

10 

9.83I?o9 

I755 

1800 

i845 

1891 

i936 

49 

10 

9.8479i3 

7957 

8002 

8o47 

8o92 

8i36 

49 

ii 

i98i 

2026 

20-72 

2117 

2162 

2208 

48 

ii 

8181 

8226 

82-70 

83i5 

836o 

84o5 

48 

12 

2253 

220.8 

2343 

2389 

2434 

2479 

47 

12 

8449 

8494 

8539 

8583 

8628 

8673 

47 

i3 

2525 

257O 

26i5 

2660 

27o6 

275l 

46 

i3 

'  8717 

8762 

88o7 

885i 

8896 

894i 

46 

i4 

2796 

2842 

2887 

2932 

2977 

3o23 

45 

i4 

8986 

9o3o 

9075 

9I2O 

9i64 

9209 

45 

i5 

3o68 

3n3 

3i58 

3204 

3249 

3294 

44 

i5 

9254 

9298 

9343 

9388 

9432 

9477 

44 

16 

3339 

3385 

343o 

3475 

3520 

3566 

43 

16 

9522 

9566 

96n 

9656 

97oo 

9745 

43 

17 

36n 

3656 

37oi 

3747 

3792 

3837 

42 

17 

979° 

9834 

9879 

9924 

9968 

..i3 

42 

18 

3882 

3927 

3973 

4oi8 

4o63 

4io8 

4i 

18 

9.85oo57 

0102 

oi47 

oi9i 

0236 

0281 

4i 

J9 

4i54 

4199 

4244 

4289 

4334 

438o 

4o 

i9 

o325 

o37o 

o4i5 

o459 

o5o4 

o548 

4o 

20 

9.  834425 

447o 

45i5 

456i 

46o6 

465  1 

39 

20 

9.85o593 

o638 

0682 

0727 

o772 

0816 

39 

21 

4696 

474i 

4787 

4832 

4877 

4922 

38 

21 

0861 

0905 

o95o 

o995 

io39 

1084 

38  , 

22 

4967 

5oi2 

5o58 

5io3 

5i48 

5i93 

37 

22 

II29 

11:73 

1218 

1262 

i3o7 

i352 

37 

23 

5238 

5284 

5329 

5374 

5419 

5464 

36 

23 

i396 

i44i 

i485 

i53o 

i575 

i6i9 

36 

24 

55o9 

5555 

56oo 

5645 

5690 

5735 

35 

24 

1  664 

I7o8 

i753 

i797 

1842 

i887 

35 

25 

6780 

5826 

587i 

59i6 

5961 

6006 

34 

25 

i93i 

i976 

2O2O 

2o65 

2IIO 

2i54 

34 

26 

6o5i 

6096 

6142 

6i87 

6232 

6277 

33 

26 

2199 

2243 

2288 

2332 

2377 

2422 

33 

27 

6322 

6367 

6412 

6458 

65o3 

6548 

32 

27 

2466 

25ll 

2555 

2600 

2644 

2689 

32 

28 

6593 

6638 

6683 

6-728 

6773 

68j9 

3i 

28 

2733 

2778 

2823 

2867 

29I2 

2956 

3i 

29 

6864 

6909 

6954 

6999 

7044 

7o89 

3o 

29 

3ooi 

3o45 

3o9o 

3i34 

3i79 

3223 

3o 

3o 

9.837i34 

7179 

7225 

7270 

73i5 

736o 

29 

3o 

9.S53268 

33i3 

3357 

3402 

3446 

349i 

29 

3i 

74o5 

745o 

7495 

7585 

763o 

28 

3i 

3535 

358o 

3624 

3669 

37i3 

3758 

28 

32 

7675 

7721 

7766 

78n 

7856 

79oi 

27 

32 

38o2 

3847 

389i 

3936 

398o 

4o25 

27 

33 

7946 

7991 

8o36 

8081 

8126 

8171 

26 

33 

4o69 

4n4 

4i58 

4247 

4292 

26 

34 

82168261 

83o7 

8352 

8397 

8442 

25 

34 

4336 

438i 

4425 

447o 

45i4 

4559 

25 

35 

84878532 

8577 

8622 

8667 

8712 

24 

35 

46o3 

4648 

4692 

4737 

478i 

4826 

24 

36 

8757 

8802 

8847 

8892 

8937 

8982 

23 

36 

487o 

49i5 

4959 

5oo4 

5o48 

5o93 

23 

37 

9027 

9072 

9117 

9.162 

920-7 

9252 

22 

37 

5i37 

5i82 

5226 

527I 

53i5 

536o 

22 

38 

9297 

9343 

^388 

9433 

9478 

9523 

21 

38 

54o4 

544g 

5493 

5537 

5582 

5626 

21 

39 

9568 

9613 

9658 

9703 

9748 

9793 

2O 

39 

567i 

57i5 

576o 

58o4 

5849 

5893 

2O 

4o 

9.839838 

9883 

9928 

9973 

..18 

..63 

I9 

4o 

9.855938 

5982 

6026 

6o7i 

6n5 

6160 

19 

4i 

9.840108 

oi53 

0198 

0243 

0288 

o333 

18 

4i 

6204 

6249 

6293 

6338 

6382 

6426 

1  8 

42 

o378 

0423 

o468 

o5i3 

o558 

o6o3 

'7 

42 

647i 

65i5 

656o 

66o4 

6649 

6693 

17 

43 

o648 

o693 

o737 

0-782 

o827 

0872 

16 

43 

6737 

6-782 

6826 

687i 

69i5 

6959 

16 

44 

091-7 

096>2 

1007 

1052 

io97 

1142 

i5 

44 

7oo4 

7o48 

7°93 

7i37 

7l82 

-7226 

i5 

45 

n87 

1232 

1277 

1322 

i367 

l4l2 

i4 

45 

7270 

73!5 

7359 

74o4 

7448 

7492 

i4 

46 

i457 

1502 

1  547 

l592 

i637 

1682 

i3 

46 

7537 

758i 

7626 

767o 

77i4 

7759 

i3 

47 

I727 

1771 

1816 

1861 

1906 

i95i 

12 

47 

78o3 

7848 

7892 

7936 

798i 

8026 

12 

48 

i996 

2086 

2l3l 

21-76 

2221 

II 

48 

8o69 

8n4 

8i58 

8203 

8247 

8291 

II 

49 

2266 

23ll 

2355 

2400 

2445 

249O 

IO 

49 

8336 

838o 

8424 

8469 

85i3 

8558 

IO 

5o 

9  842535 

258o 

2625 

26-70 

27l5 

2760 

9 

5o 

9.  858602 

8646 

869i 

8735 

8779 

8824 

9 

5i 

28o5 

2849 

2894 

2939 

2984 

3o29 

8 

5i 

8868 

89I2 

8957 

9OOI 

9o45  9o9o 

8 

52 

3o74 

3n9 

3i64 

32O9 

3253 

3298 

7 

52 

9i34 

9!78 

9223  9267 

93ii  9356 

7 

53 

3343 

3388 

3433 

3478 

3523 

3568 

6 

53 

94oo 

9444 

9489  9533 

9577  9622 

6 

54 

36i2 

3657 

3702 

3747 

3-792 

3837 

5 

54 

9666 

97io 

9755  9799 

9843  9888 

5 

55 

3882 

3927 

397i 

4oi6 

4o6i 

4io6 

4 

55 

9932 

9976 

..21  ..65 

.109  .  1  54 

4 

56 

4i5i 

4196 

424i 

4285 

433o 

4375 

3 

56 

9.86oi98 

0242 

02870331 

o375o42o 

3 

57 

4420 

4465 

45io 

4554 

4599 

4644 

2 

57 

o464 

o5o8  o553!o5o7 

o64i  o685 

2 

58 
59 

4689 
4958 

4734 
5oo3 

4779 
5o48 

4823 
5o92 

4868 
5i37 

49i3 
5i82 

I 
O 

58 

o73oo774 
o995|io4o 

0818  0862 

1084  1128 

O9o7.o95i 

II72  I2I7 

I 
o 

60" 

50"  |  40" 

30" 

20" 

10" 

a 

60"    |  50"   40"  |  30"   20"   10" 

2 

Co-tangent  of  55  Degrees. 

9 

Co-tangent  of  54  Degrees. 

I 

p  p  A  I"  2"  3"  4"  5"  6"  7"  8"  9" 
p   inl  5  9  14  18  23  27  32  36  41 

P  Part  5  l"  ~"  3"  4"  5"  6"  7"  8"  9" 
I  4   9  13  18  i2  27  31  36  40  | 

LOGARITHMIC    SINES. 


d 

c 

Sine  of  36 

Degrees. 

d 

" 

Sine  of  37  Degrees. 

51 

0" 

10" 

20" 

30' 

40"  |  50" 

0" 

10"   20" 

30" 

40" 

50" 

c 

9.769219 

9248 

9277^806 

9335  936^ 

59 

0 

9-779463 

9491 

95i9 

9547 

9575 

96o3 

59 

I 

9393 

9421 

g45o  9479 

95689537 

58 

I 

963i 

9669 

9686 

97i4 

977° 

58 

2 

9566 

9595 

96249653 

9682  97n 

67 

2 

979s 

9826 

9854 

9882 

99io 

9938 

57 

3 

974o 

9769 

9798|9827 

98569884 

56 

| 

9966 

9993 

.  .21 

..49 

••77 

.  io5 

56 

4 

,99i3 

9942 

9971 

•  .  .  . 

..29 

..58 

55 

I 

9.78oi33 

0161 

oi89 

0216 

0244 

O272 

55 

5|9.77oo87 

0116 

oi45  oi73 

02O2  O23l 

54 

1 

o3oo 

o328 

o356 

o384 

o4n 

o439 

54 

6 
7 

0260 
o433 

0289 

0462 

o3i8o347 
0491  o52o 

o376  o4o4 
o549io577 

53 

52 

6 

o467 
o634 

o495 
0662 

o523 
0600 

o55i 
0-718 

o578 
o745 

0606 
o773 

53 

52 

8 

0606 

o635 

0664  069^ 

0722 

o75o 

5i 

8 

•  0801 

0829 

o857 

0884 

O9I2 

o94o 

5i 

9 

°779 

0808 

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o895 

O923 

5o 

9 

0968 

0996 

IO23 

io5i 

io79 

II07 

5o 

IO 

9.77o952 

o98i 

1010  1039 

io67 

io96 

49 

IO 

9.78n34 

1162 

II9O 

1218 

1246 

I273 

49 

ii 

1125 

n54 

n83  i2ii 

1240 

1269 

48 

ii 

i3oi 

i329 

i357 

1  384 

l4l2 

i44o 

48 

12 

I298 

i326 

i355  i384 

i4i3 

i44i 

47 

12 

i468 

i495 

i523 

i55i 

i578 

1606 

47 

i3    1470 

i499 

1528^556 

i585 

i6i2 

46 

i3 

i634 

1662 

i689 

i7i7 

i745 

I772 

46 

i4 

1  643 

1671 

I7OO  I729 

i758 

i786 

45 

\L 

1800 

1828 

i856 

i883 

i9n 

i939 

45 

i5 

i8i5 

1  844 

i872  1901 

1930 

i959 

44 

i5 

i966 

i994 

2O22 

2049 

2077 

2105 

44 

16 

1987 

2016 

2O45  2O73 

2IO2 

2l3l 

43 

16 

2l32 

2160 

2188 

22l5 

2243 

227I 

43 

i7 

2169 

2188 

2217  2245 

2274  23o3 

42 

17 

2298 

2326 

2354 

238i 

24o9 

2437 

42 

18 

233i 

236o 

2389 

24l7 

2446  2475 

4i 

18 

2464 

2492 

2620 

2547 

2575 

2602 

4i 

'9 

25o3 

2532 

256i 

2589 

2618 

2646 

4o 

19 

263o 

2658 

2685 

27l3 

274l 

2768 

4o 

20i9.772675 

2704 

2732 

276l 

2790 

2818 

39 

20 

9.782796 

2823 

285i 

2879 

29o6 

2934 

39 

21 

2847 

2875 

2904  2933 

2961 

2990 

38 

21 

2961 

2989 

3oi7 

3o44 

3o72 

3o99 

38 

22 

3oi8 

3o47 

3o76 

3io4 

3i33 

3i6i 

37 

22 

3l27 

3i82 

32IO 

3237 

3265 

37 

23 

3190 

32i9 

3247 

3276 

33o4 

3333 

36 

23 

3292 

3320 

3347 

3375 

3402 

343o 

36 

24 

336i 

339o 

34i8 

3447 

3476  35o4 

35 

24 

3458 

3485 

35i3 

354o 

3568 

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35 

25 

3533 

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3590 

36i8 

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25 

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34 

26 

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26 

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27 

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28 

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28 

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29 

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4245 

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29 

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3o 

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29 

3o 

9-784447 

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29 

3i 

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4587 

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4672 

47oo 

28 

3i 

4612 

4639 

4667 

4694 

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4749 

28 

32 

4729 

4757 

4786!48i4 

4842 

487i 

27 

32 

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48o4 

483i 

4858 

4886 

49i3 

27 

33 

4899 

4928 

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5oi3 

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26 

33 

4941 

4968 

4995 

5o23 

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5o78 

26 

34    5o7o 

5o98 

5i26 

5i55 

5i83 

5212 

25 

34 

5io5 

5i32 

5i6o 

5i87 

52i4 

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25 

35    524o 

5268 

5297 

5325 

5353 

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24 

35 

6269 

5296 

5324 

535i 

5378 

54o6 

24 

36    54io 

5438 

5467 

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5552 

23 

36 

5433 

546i 

5488 

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23 

37    558o 

56o8 

5637 

5665 

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5722 

22 

37 

5597 

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381    575o 

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21 

38 

576i 

5788 

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42 

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6457 

6485 

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6542 

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17 

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43 

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6627 

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6739 

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43 

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44 

6768 

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6878 

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45 

6937 

6965 

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7o78 

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45 

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6960 

6987 

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7042 

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7106 

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47 

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7359 

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12 

47 

7232 

7259 

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7367 

12 

48    7444 

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7628 

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48 

7395 

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49    7613 

764i 

7669 

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7725 

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49 

7557 

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78io 

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9 

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8 

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52 

8119 

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52 

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7 

53 

8287 

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8399 

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6 

53 

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6 

54 

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8539 

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5 

54 

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8478 

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5 

55 

8624 

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8764 

4 

55 

8532 

8559 

8586 

86i3 

864o 

8667 

4 

56 

8792 

8820 

8848 

8876 

89o4 

8932 

3 

56 

8694 

872I 

8748 

8775  8802 

8829 

3 

58 

8960 
9128 

8988 
9i56 

9oi6 
9i83 

9044 
9211 

90-72 
9239 

9ioo 
9267 

2 
I 

57 
58 

8856 
9oi8 

8883 
9o45 

8910 

9072 

89378964 
9099  9126 

8991 

2 
I 

59 

9295 

9323 

935i 

9379  g4o7 

9435 

0 

59 

9180 

9207 

9234 

9261  92889815  o 

6P"    |  50" 

40" 

30"   20" 

10" 

, 

GO"     50"  |  40" 

30"  :  20"  |  10"   „. 

Co-sine  of  53 

Degrees. 

2 

Co-sine  of  52  Degrees.     a 

„  v  .<  1"  2"  3"  4" 

5"  6"  7"  8"  9" 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

r.PartJ  3   6   9  n 

14  17  20  23  26 

l.Fart^  3   5   8  11  14  16  19  22  25 

LOGARITHMIC    TANGENTS. 


Ci 


1 

1 

Tangent  of  36  Degrees. 

.3 

Tangent  of  37  Degrees. 

2 

0" 

10" 

20" 

30" 

40" 

50" 

& 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.861261 

i3o5 

i35o 

I3g4 

i438 

1482 

59 

o 

9.  877114 

7i58 

72O2 

7246 

7290 

7333 

59 

I 

i527 

1571 

1616 

i659 

i7o4 

1748 

58 

i 

7377 

7421 

7465 

75o9 

7552 

7596 

58 

•2. 

1792 

i837 

1881 

I925 

1969 

2OI^ 

$7 

2 

7640 

7684 

7728 

777i 

78i5 

7859 

57 

3 

2o58 

2IO2 

2146 

2191 

2235 

2279 

56 

3 

79°3 

7947 

799° 

8o34 

8o78 

8122 

56 

4 

2323 

2368 

2412 

2456 

25oo 

2545 

55 

4 

8i65 

8209 

8253 

8297 

834i 

8384 

55 

£ 

2689 

2633 

2677 

2721 

2766 

2810 

54 

5 

8428 

8472 

85i6 

8559 

86o3 

8647 

54 

6 

2854 

2808 

2943 

2987 

3o3i 

3075 

53 

6 

869i 

8734 

8778 

8822 

8866 

89o9 

53 

7 

SlIQ 

3i64 

32o8 

3252 

3296 

334i 

52 

7 

8953 

8997 

9o4i 

9o85 

9I28 

9I72 

52 

8 

3385 

3429 

3473 

35-17 

3562 

36o6 

5i 

8 

9216 

9260 

93o3 

9347 

939i 

9435 

5i 

9 

365o 

3694 

3738 

3783 

3827 

3871 

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9 

9478 

9522 

9566 

9609 

9653 

9697 

5o 

10 

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3959 

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4o48 

4092 

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49 

10 

9.87974i 

9784 

9828 

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9959 

49 

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4225 

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43i3 

4357 

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48 

ii 

9.88ooo3 

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12 

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47 

12 

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44 

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16 

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43 

16 

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43 

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42 

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1620 

1  664 

1-708 

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42 

18 

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18 

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20 

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6697 

6741 

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39 

20 

9.  88s363 

2406 

245o 

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39 

21 

6829 

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38 

21 

2625 

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2843 

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22 

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37 

22 

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2930 

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37 

23 

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7402 

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23 

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3279 

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3367 

36 

24 

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7667 

7711 

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35 

24 

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3454 

3498 

354i 

3585 

3628 

35 

25 

7887 

793i 

7975 

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34 

25 

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3759 

38o3 

3847 

389o 

34 

26 

8i52 

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8284 

8328 

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33 

26 

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3977 

4021 

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4i52 

33 

27 

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846o 

85o4 

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32 

27 

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4239 

4283 

4326 

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32 

28 

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3i 

28 

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4588 

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29 

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29 

4719 

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29 

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29 

3i 

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28 

3i 

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28 

32 

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978i 

9825 

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27 

32 

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33 

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26 

34 

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25 

34 

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6070 

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25 

35 

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35 

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24 

36 

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23 

36 

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21 

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1980 

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9.887594 

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18 

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18 

42 

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42 

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43 

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16 

43 

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8552 

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16 

44 

29o3 

2947 

2991 

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3i23 

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44 

8639 

8682 

8726 

8769 

88i3 

8856 

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45 

3167 

3211 

3255 

3299 

3342 

3386 

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45 

8900 

8943 

8987 

9o3o 

9o74 

9117 

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46 

343o 

3474 

35i8 

3562 

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46 

9161 

9204 

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9334 

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47 

3694 

3738 

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3825 

3869 

39i3 

12 

47 

9421 

9.465 

95o8 

9552 

9595 

9639 

12 

48 

3957 

4ooi 

4o45 

4089 

4i33 

4177 

II 

48 

9682 

9726 

9769 

98i3 

9856 

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II 

49 

4220 

4264 

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4352 

4396 

444o 

IO 

49 

9943 

9987 

..3o 

..74 

.n7 

.160 

10 

5o 

9.874484 

4528 

4572 

46i5 

4659 

4703 

9 

5o 

9.890204 

0247 

O29I 

o334 

o378 

O42I 

9 

5i 

4747 

4791 

4835 

4879 

4923 

4966 

8 

5i 

o465 

o5o8 

o552 

o595 

o639 

0682 

8 

52 

5oio 

5o54 

5098 

5i42 

5i86 

523o 

7 

52 

O725 

o769 

0812 

o856 

o899 

o943 

7 

53 

5273 

53i7 

536i 

54o5 

5449 

5493 

6 

53 

0986 

io3o 

io73 

1116 

1160 

1203 

6 

54 

5537 

558o 

5624 

5668 

5712 

5756 

5 

54 

1247 

I29O 

i334 

i377 

i4ai 

i464 

5 

55 

58oo 

5843 

5887 

SgSi 

5975 

6oi9 

4 

55 

i5o7 

i55i 

i594 

i638 

1681 

1725 

4 

56 

6o63 

6io7 

6i5o 

6194 

6238 

6282 

3 

56 

i768 

1811 

i855 

i898 

I942 

i985 

3 

57 

6326 

637o 

64i3 

6457 

65oi 

6545 

2 

5? 

2028 

2072 

2Il5 

2l59 

22O2 

2246 

a 

58 

6589 

6632 

6676 

6720 

6764 

6808 

I 

58 

2289 

2332 

2376 

24i9 

2463 

25o6 

i 

59 

6852 

6895 

6939 

6983 

7027 

7071 

O 

59 

2549 

2593 

2636 

2680 

2723 

2766 

o 

60"     50" 

40" 

30" 

20" 

10" 

d 

60"     50" 

40" 

30"   20"   10"  I  . 

Co-tangent  of  53  Degrees. 

.H 

E 

Co-tangent*  of  52  Degrees. 

9 

P  P^t  5  !"  2"  3//  4//  5"  6//  7"  8//  9"  'I  P  P  1  5  1"  2"  3"  4"  5"  6"  7"  8"  9" 

L"£  4   9  13  18  22  26  31  35  40  ||     irt}  4   9  13  17  22  26  31  35  39 

LOGARITHMIC    SINES. 


d 

Sine  of  38  Degrees. 

d 

Sine  of  39  Degrees. 

1 

* 

i> 

10" 

20"   30"  I  40" 

50" 

^     0" 

10" 

20" 

30"   40" 

50" 

0 

I 

2 

95o^ 
9665 

93699396 
953i9557 
96929719 

9423  945o 
9584  96n 
97469773 

9477 

9638 
9800 

59 

58 

0 
I 

9.798872 

9028 

9i84 

8898 
9054 
9210 

8924 
9080 

9236 

895o 
9io6 

9262 

8976 
9132 

9287 

9002 

9i58 
93i3 

59 

58 
57 

2 

9827 

9854 

9880  9907  9934 

9961 

56 

3 

9339 

9365 

939i 

94i7 

9443 

9469 

56 

4 

9988 

.  .  i5 

.  .42'.  .69  .  .96 

.  122 

55 

4 

9495 

9521 

9547 

9573 

9599 

96a5 

55 

5 

9.790149 

0176  02O3 

O23o  0267 

0284 

54 

c 

965i 

9677 

97o3 

9728 

9754 

978o 

54 

6 

o3io 

o337o364 

o39i 

o4i8 

0445 

53 

6 

98o6 

9832 

9858 

9884 

9910 

9936 

53 

7 

0471 

o498 

o525  o552  o579 

0606 

52 

7 

9962 

9987 

.  .  i3 

..39 

52 

8 

o632 

o659 

068607130740 

0767 

5i 

8 

9.  800117 

oi43 

oi69 

oi95 

O22O 

0246 

5i 

9 

-   o793 

0820 

0847 

0874  o9oi 

0927 

5o 

9 

0272 

0298 

o324 

o35o 

o376 

o4oi 

5o 

10 

9.79o954 

o98i 

1008 

io34 

1061 

1088 

49 

10 

9.  800427 

o453 

o479 

o5o5 

o53i 

o556 

49 

ii 

ni5 

1142 

1168 

iio5 

1222 

I249 

48 

ii 

o582 

0608 

o634 

0660 

0686 

0711 

48 

12 

1275 

1302 

i329 

1  356 

I  382 

i4o9 

47 

12 

o737 

o763 

o789 

o8i5 

o84o 

0866 

47 

13 

i436 

i463 

i489 

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I  543 

1570 

46 

i3 

o892 

0918 

o944 

o969 

o995 

1021 

46 

i4 

i596 

1623 

i65o 

1676 

1703 

1730 

45 

i4 

1047 

1073 

io98 

1124 

n5o 

II76 

45 

i5 

1757 

i783 

1810 

i837 

i863 

i89o 

44 

i5 

1201 

1227 

1253 

I279 

i3o5 

i33o 

44 

16 

1917 

i943 

1970 

i997 

2024 

2o5o 

43 

16 

i356 

i382 

1408 

i433 

i45g 

i485 

43 

17 

2077 

2104 

2i3o 

2157 

2184 

22IO 

42 

'7 

i5n 

i536 

i562 

1  588 

i6i3 

i639 

42 

18 

2237 

2264 

2290 

2317 

2344 

2370 

4i 

18 

i665 

1691 

I7i6 

1742 

i768 

i794 

4i 

'9 

2397 

24.23 

245o 

2477 

25o3 

253o 

4o 

i9 

i8i9 

i845 

i87i 

i896 

1922 

i948 

4o  • 

20 

9.792557 

2583 

2610 

2636 

2663 

269o 

39 

20 

9.8oi973 

1999 

2025 

2o5i 

2076 

2102 

39 

21 

2716 

2743 

2770 

2796 

2823 

2849 

38 

21 

2128 

2i53 

2179 

22O5 

2230 

2256 

38 

22 

2876 

29o3 

2929 

2956 

2982 

3oo9 

37 

22 

2282 

2307 

2333 

2359 

2384 

24lO 

37 

23 

3o35 

3o62 

3089 

3n5 

3l42 

3i68 

36 

23 

2436 

2461 

2487 

25  I  2 

2538 

2564 

36 

24 

3i95 

3222 

3248 

3275 

33oi 

3328 

35 

24 

2589 

26i5 

2641 

2666 

2692 

2718 

35 

25 

3354 

338i 

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3434 

346o 

3487 

34 

25 

2743 

2769 

2794 

2820 

2846 

2871 

34 

26 

35i4 

354o 

3567 

3593 

3620 

3646 

33 

26 

2897 

2922 

2948 

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2999 

3o25 

33 

27 

3673 

3699 

3726 

3752 

3779 

38"o5 

32 

27 

3o5o 

3o76 

3l02 

3127 

3i53 

3i78 

32 

28 

3832 

3858 

3885 

39ii 

3938 

3964 

3i 

28 

32o4 

3229 

3255 

3281 

33o6 

3332 

3i 

29 

399i 

4017 

4o44 

4070 

4097 

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3o 

29 

3357 

3383 

34o8 

3434 

3459 

3485 

3o 

3o 

9,794i5o 

41764203 

4229  4255 

4282 

29 

3o 

9.8o35n 

3536 

3562 

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36i3 

3638 

29 

3i 

4  3  08 

43354361 

4388  4414 

444  1 

28 

3i 

3664 

3689 

37i5 

374o 

3766 

379i 

28 

32 

4467 

44934520 

45464573 

4599 

27 

32 

38i7 

3842 

3868 

3893 

39i9 

3944 

27 

33 

4626 

4652 

4678 

47o5 

473i 

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26 

33 

397o 

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4o46 

4072 

4o97 

26 

34 

4784 

48io4837 

4863 

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49i6 

25 

34 

4l23 

4i48 

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4199 

4225 

25 

35 

4942 

49694995 

5022 

5o48 

5074 

24 

35 

4276 

43oi 

4327 

4352 

4377 

44o3 

24 

36 

5ioi 

5127  5i54 

5i8o 

52o6 

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23 

36 

4428 

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4479 

45o5 

453o 

4556 

23 

37 

5259 

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53i2 

5338 

5364 

539i 

22 

37 

458i 

4607 

4632 

4657 

4683 

4708 

22 

38 

54i7 

5443 

5470 

5496 

5522 

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21 

38 

4734 

4759 

4784 

48io 

4835 

486i 

21 

39 

5575 

56oi 

5628 

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568o 

57o7 

20 

39 

4886 

49I2 

4937 

4962 

4988 

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9.795733 

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5786 

58i2 

5838 

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19 

4o 

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5o64 

5o89 

5n5 

5i4o 

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4i 

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59i7 

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18 

4i 

5i9i 

52i6 

5242 

5267 

5292 

53i8 

18 

42 

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6i54 

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42 

5343 

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5394 

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5470 

17 

43 

6206 

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6338 

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43 

5495 

5520 

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5597 

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44 

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45 

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6547 

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45 

5799 

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5875 

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46 

6679 

6705 

673i 

6757 

6783 

6810 

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46 

595i 

5976 

6002 

6027 

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47 

6836 

6862 

6888 

6914 

6941 

6967 

12 

47 

6io3 

6128 

6i53 

6179 

6204 

6229 

12 

48 

6993 

7019  7045 

7072 

7098 

7124 

II 

48 

6254 

6280 

63o5 

633o 

6355 

638i 

II 

49 

7i5o 

7176 

7202 

7229 

7255 

7281 

10 

49 

64o6 

643i 

6456 

6482 

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6532 

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5o 

9.797307 

7333  7359 

7386 

7412 

7438 

9 

5o 

9.8o6557 

6583 

6608 

6633 

6658 

6684 

9 

5i 

7464 

7490 

75i6 

7542 

7569 

7595 

8 

5i 

67o9 

6734 

6759 

6785 

6810 

6835 

8 

52 

7621 

7647 

7673 

7699 

7725 

775i 

7 

52 

6860 

6885 

69n 

6936 

6961 

6986 

7 

53 

7777 

7804, 

783o 

7856 

7882 

79o8 

6 

53 

7011 

7o37 

7062 

7o87 

7112 

7i37 

6 

54 

7934 

7960 

7986 

8012 

8o38 

8o65 

5 

54 

7i63 

7i88 

7213 

7238 

7263 

7288 

5 

55 

8o9i 

81178143 

8169 

8i95 

8221 

4 

55 

?3i4 

7339 

7364 

7389 

74i4 

7439 

4 

56 

8247 

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8325 

835i 

8377 

3 

56 

7465 

749o 

75i5 

754o 

7565 

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3 

57 

84o3 

8429 

8455 

848i 

85o8 

8534 

2 

57 

76i5 

7666 

769i 

7716 

774i 

2 

58 
59 

856o 
8716 

8586:86128638 
8742  8768  8794 

8664  869o 

88208846 

I 
O 

58 
59 

7766 
7917 

7791 
7942 

7816 
7967 

7992 

7867  7892 
8017  8042 

I 
O 

60" 

50" 

40"   30" 

20"  I  10" 

. 

60" 

50"   40"  |  30"  )  20"   10" 

a 

Co-sine  of  51  Degrees. 

1 

Co-sine  of  50  Degrees. 

i 

(  1"  2"  3' 

4"  5"  6"  7"  8"  9" 

C  1"  2"  3"  4"  5"  6"  7"  8"  9" 

P.  Part  J  3  "5   g 

11  13  16  19  21  24 

P.  Fart  £  3   5   g  10  13  15  13  20  23  j 

LOGARITHMIC    TANGENTS. 


6  H 


fij 

Tangent  of  38  Degrees. 

.2 

Tangent  of  39  Degrees. 

j 

i  S 

0" 

10" 

20" 

30" 

40" 

50" 

* 

0"    |  10" 

20"   30"   40"  j  50" 

\  ~ 
o 

9.89.2810 

2853 

2897 

2940 

2983 

3027 

59 

0 

9.9o8369 

8412 

8455 

8498 

854i 

8584 

59 

I 

3o7o 

3n4 

3i57 

32OO 

3244 

3a87 

58 

i 

8628 

8671 

87i4 

8757 

8800 

8843 

58 

2 

333i 

3374 

34i7 

346  1 

35o4 

3547 

57 

2 

8886 

8929 

89-72 

9oi5 

9o58 

9IOI 

57 

3 

359i 

3634 

3678 

372I 

3764 

38o8 

56 

3 

9i44 

9187 

9230 

9273 

93i6 

9359 

56 

4 

385! 

3894 

3938 

398i 

4o25 

4o68 

55 

4 

94O2 

9445 

9488 

953i 

9574 

96l7 

55 

5 

4m 

4i55 

4198 

4241 

4285 

4328 

54 

5 

966o 

97o3 

9746 

9789 

9832 

9875 

54 

6 

4372 

44i5 

4458 

4502 

4545 

4588 

53 

6 

99i8 

9961 

...5 

..48 

.  .9i 

.i34 

53 

7 

4632 

4675 

4718 

4762 

48o5 

4848 

52 

7 

9.9ioi77 

O22O 

0263 

o3o6 

o349 

o392 

52 

8 

4892 

4935 

4979 

5O22 

5o65 

5io9 

5i 

8 

o435 

0478 

O52I 

o564 

o6o7 

o65o 

5i 

9 

5i52 

5i95 

5239 

5282 

5325 

5369 

5o 

9 

o693 

o736 

°779 

0822 

o865 

o9o8 

5o 

10 

9.8954i2 

5455 

5499 

5542 

5585 

5629 

49 

10 

9.9100.51 

0994 

io37 

1080 

1123 

1166 

49 

ii 

5672 

57i5 

5759 

58o2 

5845 

5889 

48 

ii 

I209 

1252 

1295 

i338 

i38i 

1424 

48 

12 

5932 

5975 

6oi9 

6062 

6io5 

6i49 

47 

12 

i467 

i5io 

i553 

i596 

i639 

1682 

47 

i3 

6l92 

6235 

6278 

6322 

6365 

64o8 

46 

i3 

I725 

i768 

1810 

i853 

i896 

I939 

46 

i4 

6452 

6495 

6538 

6582 

6625 

6668 

45 

i4 

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2O25 

2068 

2III 

2i54 

2197 

45 

i5 

67I2 

6755 

6798 

6842 

6885 

6928 

44 

i5 

2240 

2283 

2326 

2369 

24l2 

2455 

44 

16 

697i 

7oi5 

7o58 

7IOI 

7i45 

7i88 

43 

16 

2498 

254i 

2584 

2627 

2670 

27l3 

43 

i7 

723l 

7275 

73i8 

736i 

74o4 

7448 

42 

17 

2756 

2799 

2842 

2885 

20,28 

297I 

42 

18 

7491 

7534 

7578 

762I 

7664 

7707 

4i 

18 

3oi4 

3o57 

3ioo 

3i43 

3i85 

3228 

4i 

19 

775i 

7794 

7837 

788i 

7924 

7967 

4o 

19 

327I 

33i4 

3357 

34oo 

3443 

3486 

4o 

9.898oio 

8o54 

8o97 

8i4o 

8i83 

8227 

39 

20 

9.9i3529 

3572 

36i5 

3658 

37oi 

3744 

39 

21 

82-70 

83i3 

8357 

84oo 

8443 

8486 

38 

21 

3787 

383o 

3873 

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3959 

4ooi 

38 

22 

853o 

8573 

8616 

8659 

87o3 

8746 

37 

22 

4o44 

4o87 

4i3o 

4i73 

4216 

4259 

37 

23 

8789 

8832 

8876 

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8962 

9oo5 

36 

23 

4302 

4345 

4388 

443  1 

4474 

45i7 

36 

24 

9049 

9092 

9i35 

9i78 

9222 

9265 

35 

24 

456o 

46o3 

4645 

4688 

473i 

4774 

35 

25 

9308 

935i 

9395 

9438 

948i 

9524 

34 

25 

48i7 

486o 

49o3 

4946 

4989 

5o32 

34 

26 

9568 

96n 

9654 

9697 

974i 

9784 

33 

26 

5o75 

5n8 

5i6i 

52o3 

5246 

5289 

33 

27 

0827 

0870 

C\C\  T  /I 

43 

32 

27 

5332 

5375 

54i8 

546i 

55o4 

5547 

32 

—  / 
28 

v  / 
9.9Ooo87 

7   fw 

oi3o 

0173 

0216 

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o3o3 

3i 

•*  / 
28 

559o 

<j>j  j<j 
5633 

5675 

57i8 

576i 

^/^n  j 
58o4 

3i 

29 

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o389 

0432 

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3o 

29 

5847 

5890 

5933 

5976 

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6062 

3o 

3o 

9.900605 

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0821 

29 

3o 

9.916104 

6i47 

6190 

6233 

6276 

63i9 

29 

3i 

0864 

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1037 

1081 

28 

3i 

6362 

64o5 

6448 

649i 

6533 

6576 

2$ 

32 

1124 

n67 

I2IO 

1253 

I297 

i34o 

27 

32 

66i9 

6662 

67o5 

6748 

670.1 

6834 

27 

33 

i383 

1426 

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i556 

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26 

33 

6877 

6919 

6962 

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7048 

709i 

26 

34 

1642 

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I729 

I772 

i8i5 

i858 

25 

34 

7i34 

7i77 

722O 

7262 

73o5 

7348 

25 

35 

1901 

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i988 

2031 

2074 

2II7 

24 

35 

739i 

7434 

7477 

7520 

7563 

7605 

24 

36 

2160 

22O4 

2247 

229O 

2333 

2376 

23 

36 

7648 

769J 

7734 

7777 

7820 

7863 

23 

37 

2420 

2463 

25o6 

2549 

2592 

2635 

22 

37 

79o6 

7948 

799  x 

8o34 

8o77 

8120 

22 

38 

2679 

2722 

2765 

2808 

285i 

2894 

21 

38 

8i63 

8206 

8248 

8291 

8334 

8377 

21 

39 

2938 

298i 

3o24 

3o67 

3no 

3i53 

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39 

8420 

8463 

85o6 

8548 

859i 

8634 

20 

4o 

9.903197 

3240 

3283 

3326 

3369 

34l2 

19 

4o 

0.0.18677 

8720 

8763 

88o5 

8848 

8891 

19 

4i 

3456 

3499 

3542 

3585 

3628 

367i 

18 

4i 

8934 

8977 

9020 

9063 

9io5 

9i48 

18 

42 

37i4 

3758 

38oi 

3844 

3887 

SgSo 

17 

42 

9i9i 

9234 

9277 

9320 

9362 

94o5 

i7 

43 

3973 

4oi6 

4o6o 

4io3 

4i46 

4189 

16 

43 

9448 

9491 

9534 

9577 

96i9 

9662 

16 

44 

4232 

4275 

43i8 

4362 

44o5 

4448 

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44 

97o5 

9748 

9791 

9834 

9876 

99i9 

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45 

4491 

4534 

4577 

4620 

4663 

4707 

i4 

45 

9962 

...5 

.  48 

.  .9i 

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.i76 

i4 

46 

475o 

4793 

4836 

4879 

4922 

4965 

i3 

46 

9.9202I9 

0262 

o3o5 

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47 

5oo8 

5o52 

5o95 

5i38 

5i8i 

5224 

12 

47 

0476 

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o6o4 

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o69o 

12 

48 

5267 

53io 

5353 

5397 

544o 

5483 

II 

48 

0733 

o776 

08  1  9 

0861 

0904 

o947 

II 

49 

5526 

5569 

56i2 

5655 

5698 

574i 

10 

49 

o99o 

io33 

io75 

1118 

1161 

I2O4 

10 

5o 

9.905785 

5828 

587i 

59i4 

5957 

6000 

9 

5o 

9.921247 

1289 

i332 

i375 

i4i8 

i46i 

9 

5i 

6o43 

6086 

6129 

61-72 

6216 

6259 

8 

5i 

i5o3 

1  546 

i589 

i632 

i'675 

i7i7 

8 

52 

63o2 

6345 

6388 

643  1 

6474 

65i7 

7 

52 

1760 

i8o3 

1  846 

i889 

i93i 

i974 

7 

53 

656o 

66o3 

6646 

669o 

6733 

6776 

6 

53 

2017 

2060 

2103 

2i45 

2188 

223l 

6 

54 

6819 

6862 

6905 

6948 

699i 

7034 

5 

54 

2274 

23i6 

2359 

2402 

2445 

2488 

5 

55 

7077 

7120 

7i63 

7207 

725o 

7293 

4 

55 

253o 

2573 

2616 

2659 

2702 

2744 

4 

56 

?336 

7379 

7422 

7465 

75o8 

755i 

3 

56 

2787 

283o 

2873 

29l5 

2958 

3ooi 

3 

57 

7594 

7637 

768o 

7723 

7766 

7809 

2 

57 

3o44 

3o87 

3i29 

3l72 

32i5 

3258 

a 

58 

7853 

7896 

7939 

7982 

8o25 

8068 

I 

58 

33oo 

3343 

3386 

3429 

347i 

35i4 

i 

59 

8111 

8i54 

8i97 

8240 

8283 

8326 

O 

59 

3557 

36oo 

3642 

3685 

3728 

377i 

0 

60" 

50" 

40" 

30" 

20-'' 

10" 

j 

60" 

50" 

40" 

30" 

20" 

10" 

~T~ 

Co-  tangent  of  51  Degrees. 

§ 

Co-tangent  of  50  Degrees. 

§ 

p  p  A  I"  2"  3"  4"  5"  6"  7"  8"  9" 

P  Part  5  l"  2"  3//  4//  5"  6"  7"  8</  9" 

irl}  4   9  13  17  22  26  30  35  39 

r  }  4   9  13  17  21  26  30  34  39 

LOGARITHMIC    SINES. 


a      Sine  of  40  Degrees. 

o 

JH 

Sine  of  41  Degrees. 

§ 

0" 

10" 

20"  !  30" 

40" 

50" 

3 

0" 

10" 

20" 

30" 

40" 

50" 

o 

9.808067 

8o93 

8ll88l43 

8168 

8i93 

59 

o 

9.8i6943 

6967 

699i 

7016 

7040 

7064 

59 

i 

8218 

8243 

8268  8293 

83i8 

8343 

58 

i 

7088 

7112 

7i37 

7161 

7i85 

72O9 

58 

2 

8368 

8393 

84i98444 

8469 

8494 

57 

2 

7233 

7258 

7282 

73o6 

733o 

7354 

57 

3 

85i9 

8544  8569i8594  8619 

8644 

56 

r 

7379 

74o3 

7427 

745  1 

7475 

7499 

56 

4 

8669 

8694 

87i98744 

8769 

8794 

55 

L 

7524 

7548 

7572 

7596 

7620 

7644 

55 

c 

8819 

884488698894 

8919 

8944 

54 

t 

7668 

7693 

7717 

774i 

7765 

7789 

54 

6 

8969 

8994 

9019  9044 

9069 

9094 

53 

6 

78i3 

7837 

7862 

7886 

7910 

7934 

53 

7 
8 

9119 
9269 

91449169  9i9492i9 
9294'93i9!93449369 

9^244 
9394 

52 

5i 

8 

7958 
8io3 

7982 
8127 

8006 
8i5i 

8o3o 

8i75 

8o55 
8199 

8o79 
8223 

52 

5i 

9 

9419 

9444 

9469  9494 

95i9 

9544 

5o 

9 

8247 

8272 

8296 

8320 

8344 

8368 

5o 

10 

9.809569 

95949619  9643 

9668 

9693 

49 

10 

9.8i8392 

84i6 

844o 

8464 

8488 

85i2 

49 

ii 

9718 

97439768:9793 

9818 

9843 

48 

ii 

8536 

856o 

8584 

8609 

8633 

8657 

48 

12 

9868 

9893  9918  9943 

9967 

9992 

47 

12 

8681 

87o5 

8729 

8753 

8777 

8801 

47 

i3 

9.810017 

0042  006710092 

0117 

Ol42 

46 

i3 

8825 

8849 

8873 

8897 

8921 

8945 

46 

i4 

0167 

0191 

0216  0241 

0266 

029I 

45 

i4 

8969 

8993 

9oi7 

9o4i 

9o65 

9o89 

45 

i5 

o3i6 

o34i 

o366io39o 

o4i5 

o44o 

44 

i5 

9113 

9i37 

9i6i 

9i85 

9209 

9233 

44 

16 

o465 

0490  o5i5  o54o 

o564 

o589 

43 

16 

9257 

9281 

93o5 

9329 

9353 

9377 

43 

i? 

0614 

o639 

o664 

0689 

0713 

o738 

42 

*7 

9401 

9425 

9449 

9473 

9497 

952I 

42 

18 

0763 

0788 

o8i3 

o838 

0862 

0887 

4i 

18 

9545 

9569 

9593 

9617 

964i 

9665 

4i 

J9 

0912 

0937:0962 

0986 

IOII 

io36 

4o 

*9 

9689 

97i3 

9737 

976i 

9785 

9809 

4o 

20 

9  811061 

1086 

IIIO 

n35 

1160 

n85 

39 

20 

9.819832 

9856 

988o 

9904 

9928 

9952 

39 

21 

I2IO 

1234 

1259 

1284 

i3o9 

i334 

38 

21 

9976 

.  .  .  . 

..24 

..48 

..72 

..96 

38 

22 

i358 

i383 

i4o8 

i433 

i457 

1482 

37 

22 

9.  820120 

oi43 

0167 

oi9i 

02I5 

0239 

37 

23 

1507 

i532 

i556 

i58i 

1606 

i63i 

36 

23 

0263 

0287 

o3n 

o335 

o359 

o382 

36 

24 

i655 

1680' 

1705 

i73o 

1754 

i779 

35 

24 

o4o6 

o43o 

o454 

o478 

o5o2 

o526 

35 

25 

1804 

1828 

i853 

1878 

I9o3 

I927 

34 

25 

o55o 

o573 

o597 

0621 

0645 

0669 

34 

26 

1952 

J977! 

2OOI 

2026 

2o5i 

2076 

33 

26 

o693 

0717 

0740 

0764 

0788 

0812 

33 

27 

2IOO 

2125 

2i5o 

2174 

2199  2224 

32 

27 

o836 

0860 

o883 

0907 

o93i 

o955 

32 

o 
2o 

2248 

2273 

2298 

2322 

2347 

2372 

3i 

28 

°979 

ioo3 

1026 

io5o 

1074 

io98 

3i 

29 

2396 

2421 

2446 

2470 

2495 

252O 

3o 

29 

II2'2 

n46 

n69 

n93 

1217 

1241 

3o 

3o 

Q.  812544 

2569 

2594 

26l8 

2643 

2668 

29 

3o 

9.  821265 

1288 

l3l2 

i336 

i36o 

1  384 

29 

3i 

2692 

2717 

2742 

2766 

279I 

28i5 

28 

3i 

1407 

i43i 

i455 

1479 

i5o2 

i526 

28 

32 

2840 

2865  2889 

29l4 

2939 

2963 

27 

32 

i55o 

i574 

i598 

1621 

1645 

i669 

27 

33 

2988 

3oi2  3o37 

3o62 

3o86 

3m 

26 

33 

i693 

1716 

1740 

1764 

1788 

1811 

26 

34 

3i35 

3i6o3i85 

3209 

3234 

3258 

25 

34 

i835 

i859 

i883 

1906 

i93o 

i954 

2D 

35 

3283 

33073332 

3357 

338i 

34o6 

24 

35 

i977 

2OOI 

2025 

2049 

2072 

2096 

24 

36 

343o 

3455, 

3479 

35o4 

3529 

3553 

23 

36 

2120 

2i44 

2167 

2191 

22l5 

2238 

23 

37 

3578 

36o2 

3627 

365i 

3676 

3700 

22 

37 

2262 

2286 

23o9 

2333 

2357 

238i 

22 

38 

3725 

37493774 

3799 

3823 

3848 

21 

38 

2404 

2428 

2452 

2475 

2499 

2523 

21 

39 

3872 

3897  392i 

3946 

3970 

3995 

2O 

39 

2546 

2570 

2594 

2617 

264l 

2665 

20 

4o 

9.814019 

4o444o68 

4093 

4117 

4i42 

J9 

4o 

9.  822688 

2712 

2736 

2759 

2783 

2807 

J9 

4i 

4i66 

4i9i 

42i5 

4240 

4264 

4289 

18 

4i 

283o 

2854 

2878 

2901 

2925 

2948 

18 

4s 

43i3 

43384362 

4387 

44n 

4436 

17 

42 

2972 

2996 

3oi9 

3o43 

3o67 

3o9o 

17 

43 

446o 

4484 

45oo 

4533 

4558 

4582 

16 

43 

3n4 

3i37 

3i6i 

3i85 

3208 

3232 

1.6 

44 

4607 

463i  4656 

468o 

4704 

4729 

i5 

44 

3255 

3279 

33o3 

3326 

335o 

3373 

i5 

45 

4753 

47784802 

4827 

485x4876 

i4 

45 

3397 

3421 

3444 

3468 

349i 

35i5 

i4 

46 

4900 

4924  4949 

4973 

4998,5022 

i3 

46 

3539 

3562 

3586 

3609 

3633 

3656 

i3 

47 

5o46 

5071  5o95 

5l20 

5i44!5i68 

12 

47 

368o 

3704 

3727 

375i 

3774 

3798 

12 

48 

5i93 

5217 

5242 

5266 

529o'53i5 

II 

48 

382i 

3845 

3868 

3892 

39i5 

3939 

II 

49 

5339 

5364 

5388 

54i2 

5437546i 

10 

49 

3963 

3986 

4oio 

4o33 

4o57 

4o8o 

IO 

5o 

9.8i5485 

55io 

5534 

5558 

55835607 

9 

5o 

9.  824104 

4127 

4i5i 

4174 

4i98 

4221 

9 

5i 

5632 

5656568o 

57o5 

5729  5753 

8 

5i 

4245 

4268 

4292 

43i5 

4339 

4362 

8 

52 

5778 

58o2  5826 

585i 

58755899 

7 

52 

4386 

44o9 

4433 

4456 

448o 

45o3 

7 

53 

5924 

5948  5972 

5996 

6021  6o45 

6 

53 

4527 

455o 

4574 

4597 

4621 

4644 

6 

54 

6o69 

6094.6118 

6142 

6167  6191 

5 

54 

4668 

4691 

47i5 

4738 

4761 

4?85 

5 

55 

62i5 

6240  6264 

6288 

63i2  6337 

4 

55 

4808 

4832 

4855 

4879 

4902 

4926 

4 

56 
57 

636i 
6507 

6385  64o9  6434 
653i  6555  6579 

6458  6482 
66o4  6628 

3 

2 

56 
57 

4949 
5o9o 

4972 
5n3 

4996 
5i36 

5019 
5i6o 

5o43 
5i83 

5o66 

5207 

3 

2 

58 

6652 

6676  6701 

6725 

67496773 

I 

58 

523o 

5254 

5277 

53oo 

5324 

5347 

I 

59 

6798 

6822  6846  6870  6894  69i9 

O 

59 

537i 

5394 

5417 

544i 

<464 

5488 

0 

60" 

50" 

40"   30"   20"  ,  10" 

a 

60" 

50" 

40" 

30" 

20" 

10" 

a 

Co-sine  of  49  Degrees. 

a 

Co-sine  of  48  Degrees. 

s 

P  Part  5  l"  ~"  3" 

4"  5"  6"  7"  8"  .9" 

C  l"  o"  3"  4"  5"  6"  7"  8"  9" 

1  ni  2   5   7 

10  12  15  17  20  22 

1.  J  art*  2  5   7  1Q  12  14  1?  19  21 
»                  • 

LOGARITHMIC    T  A  N  G  E  N  T  y. 


_g 

Tangent  of  40  Degrees. 

B 

Tangent  of  41  Degrees. 

'S 

0" 

10" 

20"  |  30"   40"   50" 

2 

0" 

10"  |  20" 

30" 

40"  1  50"  ; 

O 

9  ,923814 

3856 

3899 

3942 

3985 

4027 

59 

o 

9.939163 

9206 

9248 

9291 

9333)9376 

59 

I 

4070 

4n3 

4i56 

4198 

424i 

4284 

58 

i 

94i8 

946l 

95o3 

9546 

95889631 

58 

2 

3 

4583 

4369 
4626 

44i2 
4669 

4455 
4711 

4498 
4754 

454o 
4797 

57 
56 

2 

3 

9673 
9928 

9716 
9971 

9758 

9801 
..56 

9843  9886 

57 
56 

4 

484o 

4882'4925 

4968 

Son 

5o53 

55 

4 

9.940183 

0226 

0268 

o3n 

o354 

o396 

55 

5 

5096 

5i39 

6181 

5224 

5267 

53io 

54 

5 

0439 

o48i 

o524 

o566 

0609 

o65i 

54 

6 

5352 

5395 

5438 

548  1 

5523 

5566 

53 

6 

0694 

o736 

o779 

0821 

0864 

o9o6 

53 

7 

5609 

5652 

5694 

5737 

578o 

5822 

52 

7 

0949 

0991 

io34 

1076 

1119 

1161 

52 

8 

5865 

5908 

595i 

5993 

6o36 

6079 

5i 

8 

I2O4 

1246 

I289 

i33i 

i374 

i4i6 

5i 

9 

6122 

6i64 

6207 

625o 

6292 

6335 

5o 

9 

i459 

i5oi 

1  544 

i586 

1628 

i67i 

5o 

10 

9,926378 

6421 

6463 

65o6 

6549 

659i 

49 

IO 

9.94i7i3 

1756 

1798 

i84i 

i883 

I926 

49 

ii 

6634 

6677 

6720 

6762 

68o5 

6848 

48 

ii 

i968 

2011 

2o53 

2096 

2i38 

2181 

48 

12 

6890 

6933 

6976 

7OI9 

7061 

7104 

47 

12 

2223 

2266 

23o8 

235i 

2393 

2436 

47 

i3 

7147 

7189 

7232 

7275 

73i7 

736o 

46 

i3 

2478 

2521 

2563 

2606 

2648 

269I 

46 

i4 

74o3 

7446 

7488 

753i 

7574 

7616 

45 

i4 

2733 

2776 

2818 

2861 

2903 

2945 

45 

i5 

7669 

7702 

7744 

7787 

783o 

7872 

44 

i5 

2988 

3o3o 

3o73 

3n5 

3i58 

320O 

44 

!6 

79i5 

7958 

8001 

8o43 

8086 

8129 

43 

16 

3243 

3285 

3328 

3370 

34i3 

3455 

43 

17 

8i7i 

8214 

8257 

8299 

8342 

8385 

42 

i7 

3498 

354o 

3583 

3625 

3667 

3710 

42 

18 

8427 

8470 

85i3 

8555 

8598 

864i 

4i 

18 

3752 

3795 

3837 

388o 

3922 

3965 

4i 

19 

8684 

8726 

8769 

8812 

8854 

8897 

4o 

J9 

4oo7 

4o5o 

4092 

4i35 

4177 

4219 

4o 

20 

9.92894o 

8982 

9025 

9068 

9110 

9i53 

39 

20 

9.944262 

43o4 

4347 

4389 

4432 

4474 

39 

21 

9i96 

9238 

9281 

9324 

9'366 

9409 

38 

21 

45i7 

4559 

4602 

4644 

4686 

4729 

38 

22 

9452 

9494 

9537 

958o 

9622 

9665 

37 

22 

4771 

48i4 

4856 

4899 

4941 

4984 

37 

23 

9708 

975o 

9793 

9836 

9878 

9921 

36 

23 

5o26 

5069 

5m 

5i53 

5i96 

5238 

36 

24 

9964 

...6 

..49 

..92 

.134 

.177 

35 

24 

5281 

5323 

5366 

54o8 

545i 

5493 

35 

25 

9.930220 

0262 

o3o5 

o348 

oSgo 

o433 

34 

25 

5535 

5578 

562O 

5663 

5705 

5748 

34 

26 

0475 

o5i8 

o56i 

o6o3 

0646 

0689 

33 

26 

579o 

5832 

5875 

59i7 

5960 

6002 

33 

27 

0731 

0774 

o8i7 

o859 

0902 

o945 

32 

27 

6o45 

6087 

6i3o 

6172 

6214 

6257 

32 

28 

0987 

io3o 

io73 

in5 

n58 

1  200 

3i 

28 

6299 

6342 

6384 

6427 

6469 

65n 

3i 

29 

1243 

1286 

i328 

i37i 

i4i4 

i456 

3o 

29 

6554 

6596 

6639 

6681 

6724 

6766 

3o 

3o 

9.931499 

1  542 

i584 

l627 

1669 

1712 

29 

3o 

9.9468o8 

685i 

6893 

6936 

6978 

7021 

2O 

3i 

i755 

1797 

i84o 

i883 

1925 

1968 

28 

3i 

7o63 

7io5 

7148 

7190 

7233 

7275 

2O 

32 

2010 

2o53 

2096 

2i38 

2181 

2224 

27 

32 

73i8 

736o 

7402 

7445 

7487 

753o 

27 

33 

2266 

2309 

235i 

2394 

2437 

2479 

26 

33 

7572 

7614 

7657 

7699 

7784 

26 

34 

2522 

2565 

26o7 

265o 

2692 

2735 

25 

34 

7827 

7869 

7911 

7954 

•7096 

8039 

25 

35 

2778 

2820 

2863 

29o6 

2948 

299I 

24 

35 

8081 

8i23 

8166 

8208 

825i 

8293 

24 

36 

3o33 

3o76 

3n9 

3i6i 

32o4 

3246 

23 

36 

8335 

8378 

8420 

8463 

85o5 

8548 

23 

37 

3289 

3332 

3374 

34i7 

3459 

35o2 

22 

37 

8590 

8632 

8675 

87i7 

8760 

8802 

22 

38 

3545 

3587 

363o 

3672 

37i5 

3758 

21 

38 

8844 

8887 

8929 

8972 

9014 

9o56 

21 

39 

38oo 

3843 

3885 

3928 

397i 

4oi3 

2O 

39 

9099 

9141 

9i84 

9226 

9268 

93n 

2O 

4o 

9.934o56 

4o98 

4i4i 

4i84 

4226 

4269 

1  9 

4o 

9.949353 

9396 

9438 

948o 

9623 

9565 

19 

4i 

43ii 

4354 

4397 

4439 

4482 

4524 

18 

4i 

96o8 

965o 

9692 

9735 

9777 

9819 

18 

42 

4567 

46  1  o 

4652 

4695 

4737 

4780 

Z7 

42 

9862 

99o4 

9989 

.-74 

J7 

43 

4822 

4865 

4908 

495o 

4993 

5o35 

16 

43 

9.95on6 

oi59 

0201 

0243 

6286 

o328 

16 

44 

5o78 

5l2I 

5i63 

5206 

5248 

5291 

i5 

44 

0371 

o4i3 

o455 

0498 

o54o 

o582 

i5 

45 

5333 

5376 

54i9 

546i 

55o4 

5546 

i4 

45 

0625 

0667 

0710 

0752 

0-794 

o837 

i4 

46 

5589 

5632 

5674 

5717 

5759 

58o2 

i3 

46 

o879 

0921 

0964 

1006 

1049 

1091 

i3 

47 

5844 

5887 

5930 

5972 

6oi5 

6057 

12 

47 

n33 

1176 

1218 

1261 

i3o3 

1  345 

12 

48 

6100 

6142 

6i85 

6227 

6270 

63i3 

II 

48 

i388 

i43o 

1472 

i5i5 

i557 

1600 

II 

49 

6355 

6398 

644o 

6483 

6525 

6568 

IO 

49 

1  642 

i684 

1727 

1769 

i8n|i854 

IO 

5o 

9.936611 

6653 

6696 

6738 

6781 

6823 

9 

5o 

9.951896 

i938 

1981 

2023 

20662108 

9 

5i 

6866 

6908 

695i 

6994 

7o36 

7079 

8 

5i 

2i5o 

2193 

2235 

2277 

2320  2362 

8 

52 

7121 

7i64 

7206 

7249 

729i 

7334 

7 

52 

24o5 

2447 

2489 

2532 

2574 

2616 

7 

53 

7377 

74i9 

7462 

75o4 

7547 

7589 

6 

53 

2659 

2701 

2743 

2786 

2828 

2870 

6 

54 

7632 

7674 

7717 

7759 

7802 

7845 

5 

54 

29i3 

2955 

2998 

3o4o 

3o82 

3i25 

5 

55 
56 

7887 
8142 

793o 

8!85 

7972 
8227 

8oi5 
8270 

8o57 
83i2 

8100 
8355 

4 
3 

55 
56 

3i67 
342i 

3209(3252 
3463  35o6 

3294 

3548 

3336:3379 
35qi  3633 

4 
3 

58 

8398 
8653 

844o 
8695 

8483 
8738 

8525 
8780 

8568 
8823 

8610 

8865 

2 
I 

58 

367537i8376o 
3929  3972  4oi4 

38o2  3845  3887 
4o56!4o99!4i4i 

2 
I 

5q 

8908  8950 

8993 

9o35 

9.078 

9121 

O 

59 

4i83  4226  4268  43io'4353;4395 

O 

60"     50"   40" 

30"   20" 

10" 

d 

60"     50"   40"   30"   20'   10" 

-• 

1   |   Co-tangent  of  49  Degrees. 

Co-tangent  of  48  Degrees. 

1 

1  P  P-rtJ  1"  2"  3"  4"  5"  6"  7"  8"  9" 
I  4   9  13  17  21  26  30  34  38 

P  PnrtJ  l"  2"   3"  4//  5"  6"  7>'  8"  9" 

I  4   8  13  17  21  25  30  34  38 

LOGARITHMIC    SINES. 


I 

Sine  of  42  Degrees. 

d 

Sine  of  43  Degrees. 

Ii 

0" 

10" 

20" 

30" 

40"   50" 

§ 

0" 

10"  |  20" 

30" 

40" 

50" 

o 

9.825511 

5534 

5558, 

558i 

56o4  5628  59 

0 

9.833783 

38o6  3828 

385i 

3874 

3896 

59 

I 

2 

565i 
6791 

5675 
58i5 

5698, 

5838' 

572I 

586i 

5745 
5885 

576858 
5908157 

I 

2 

39i9 
4o54 

4077 

3964 
4o99 

3986 

4l22 

4oo9 
4i44 

4o32 
4167 

58 
57 

3 

593i 

5955 

5978  6001 

6o25  6o48 

56 

3 

4i89 

4212 

4234 

4257 

4280 

43o2 

56 

4 

6071 

6095 

6118  6i4i 

6i65 

6188 

55 

4 

4325 

4347 

437o 

4392 

44i5 

4437 

55 

5 

6211 

6235 

62586281 

63o5 

6328 

54 

5 

446o 

4482 

45o5 

152J 

455o 

4572 

54 

6 

635i 

6375 

6398'642i 

6444 

6468 

53 

6 

4595 

4617 

464o 

i662 

4685 

4707 

53 

7 

6491 

65i4 

6538656i 

6584 

6607 

52 

7 

4?3o 

4?52 

4775 

4797 

4820 

4842 

52 

8 

663i 

6654 

6677 

67oi 

6724 

6747 

5i 

8 

4865 

4887 

49io 

^932 

4954 

4977 

5i 

9 

6770 

6794 

68i7,684o 

6863 

6887 

5o 

9 

4999 

5O22 

5o44 

5067 

5o89 

5lI2 

5o 

10 

9.826910 

6933 

6956  698o 

7oo3 

7026 

49 

IO 

9.835i34 

5i57 

5i79 

52OI 

5224 

5246 

49 

ii 

7049 

7o73 

7o96 

7119 

7142 

7i65 

48 

ii 

5269 

5291 

53i4 

5336 

5358 

538i 

48 

12 

7189 

7212 

7235' 

7258 

7282 

73o5 

47 

12 

54o3 

5436 

5448 

547i 

5493 

55i5 

47 

i3 

7328 

735i 

7374 

7398 

7421 

7444 

46 

i3 

5538 

556o 

5583 

56o5 

5627 

565o 

46 

i4 

7467 

7490 

?5j4 

7537 

756o 

7583 

45 

i4 

5672 

5695 

57i7 

5739 

5762 

5784 

45 

1  5 

7606 

7629 

7653, 

7676 

7699 

7722 

44 

i5 

58o7 

5829 

585i 

5874 

5896 

59i8 

44 

16 

7745 

7768 

7792 

78i5 

7838 

7861 

43 

16 

594i 

5963 

5986 

6008 

6o3o 

6o53 

43 

I7 

7884 

7907 

793i 

7954 

7977 

8000 

42 

i7 

6o75 

6o97 

6120 

6142 

6i64 

6187 

42 

18 

8023 

8o46 

8o69  8o93 

8116 

8i39 

4i 

18 

6209 

623i 

6254 

6276 

6298 

632i 

4i 

19 

8162 

8i85 

8208  823i 

8254 

8278 

4o 

J9 

6343 

6365 

6388 

64io 

6432 

6455 

4o 

20 

9.828301 

8324 

8347 

837o 

8393 

84i6 

39 

20 

9.  836477 

6499 

6522 

6544 

6566 

6589 

39 

21 

8439 

8462 

8485 

85o9 

8532 

8555 

38 

21 

6611 

6633 

6656 

6678 

67oo 

6722 

38 

22 

8578 

8601 

86248647 

8670 

8693 

37 

22 

6745 

6767 

6789 

6812 

6834 

6856 

37 

23 

8716 

8739 

8762 

8786 

8809 

8832 

36 

23 

6878 

69oi 

6923 

6945 

6968 

699° 

36 

24 

8855 

8878 

89oi 

8924 

8947 

8970 

35 

24 

7012 

7o34 

7o57 

7°79 

7101 

7123 

35 

25 

8993 

9016 

9o39  9o62 

9o85 

9108 

34 

25 

7i46 

7168 

7i9o 

7212 

7235 

7257 

34 

26 

9i3i 

9i54 

9i77  92oo  9223 

9246 

33 

26 

7279 

73oi 

7324 

7346 

7368 

739° 

33 

27 

9269 

9292 

93i5 

9338 

936i 

9384 

32 

27 

7412 

7435 

7457 

7479 

75oi 

7524 

32 

28 

9407 

943o 

9453 

9476 

9499 

9522 

3i 

28 

7546 

7568 

759o 

7612 

7635 

7657 

3i 

29 

9545 

9568 

95qi 

96i4 

9637 

9660 

3o 

29 

7679 

7701 

7723 

7746 

7768 

779° 

3o 

3o 

9.829683 

97o6 

9729|9752 

9776 

9798 

29 

3o 

9.  837812 

7834 

7857 

7879 

79oi 

7923 

29 

3i 

9821 

9844 

9867  9890 

99i3 

9936 

28 

3i 

7945 

7967 

799° 

8012 

8o34 

8o56 

28 

32 

995o 

9982 

...5 

..28 

..74 

27 

32 

8078 

8100 

8i23 

8i45 

8167 

8i89 

27 

339.830097 

0120 

0142  oi65 

0188 

02  1  1 

26 

33 

8211 

8233 

8256 

8278 

83oo 

8322 

26 

34l     0234 

O257 

0280  o3o3 

o326 

o349 

25 

34 

8344 

8366 

8388 

84io 

8433 

8455 

25 

35 

0372 

o395 

0417 

o44o 

o463 

o486 

24 

35 

8477 

8499 

852i 

8543 

8565 

8587 

24 

^6 

oSog 

o532 

o555 

o578 

0601 

0624 

23 

36 

8610 

8632 

8654 

8676 

8698 

8720 

23 

37 

o646 

o669 

0692 

0715 

o738 

0761 

22 

37 

8742 

8764 

8786 

8808 

883i 

8853 

22 

38 

o784 

0807 

08290852 

0875 

o89S 

•21 

38 

8875 

8897 

89i9 

894i 

8963 

8985 

21 

39 

0921 

o944 

0967 

°989 

1012 

io35 

20 

39 

9007 

9029 

9o5i 

9°73 

9o95 

9n8 

2O 

4o 

9.83io58 

io8i 

uo4 

1127 

1149 

1172 

I9 

4o 

9.839140 

9l62 

9184 

92o6 

9228 

925o 

J9 

4i 

1195 

1218 

1241 

1263 

1286 

i3o9 

18 

4i 

9272 

9294 

93i6 

9338 

936o 

9382 

18 

42 

i332 

i355 

i378 

i4oo 

1423 

1  446 

17 

42 

9404 

9426 

9448 

9470 

9492 

95i4 

i» 

43 

i469 

l492 

i5i4 

i537 

i56o 

i583 

16 

43 

9536 

9558 

958o 

96o2 

9624 

9646 

if 

44 

1606 

1628 

i65i 

i674 

1697 

1720 

i5 

44 

9668 

969o 

9-712 

9734 

9756 

9778 

i5 

45 

1742 

1765 

1788 

1811 

i833 

!856 

i4 

45 

9800 

9822 

9844 

9866 

9888 

99io 

i4 

46 

1879 

I902 

1924 

i947 

1970 

i993 

i3 

46 

9932 

9954 

9976 

9998 

.  .20 

..42 

i3 

47 

20l5 

2o38 

2061 

2084 

2106 

2I29 

12 

47 

9.840064 

0086 

0108 

oi3o 

Ol52 

OI7^ 

12 

48 

2l52 

2175 

2197 

2220 

2243 

2266 

II 

48 

0196 

0218 

0240 

0262 

0284 

o3o6 

ii 

49 

2288 

23ll 

2334 

2356 

2379 

2402 

IO 

49 

o328 

o35o 

0372 

o393 

o4i5 

o437 

IO 

5o 

9.832425 

2447 

2470 

2493 

25i5 

2538 

9 

5o 

9.84o459 

o48i 

o5o3 

o525 

o54? 

o569 

9 

5i 

256i 

2584 

2606 

2629 

2652 

2674 

8 

5i 

o59i 

06  1  2 

o635 

o657 

0678 

o7oo 

8 

52 

2697 

2720 

2742 

2765 

2788 

2810 

7 

52 

0722 

o744 

0766 

0788 

0810 

o832 

7 

53 

2833 

2856 

2878 

29OI 

2924 

2946 

6 

53 

o854 

o876 

0897 

o9i9 

o94i 

096!: 

6 

54 

2969 

2992- 

3oi4 

3o37 

3o6o 

3082 

5 

54 

o985 

IOO7 

1029 

io5i 

1072 

109^ 

5 

55 

3io5 

3i28 

3i5o 

3I73 

3196 

32i8 

4 

55 

1116 

n38 

1160 

1182 

120^ 

1226 

4 

56    3241 

3263 

3286 

33o9 

333i 

3354 

3 

56 

1247 

I269 

1291 

i3i3 

i335 

!357 

3  j 

57    3377 

3399 

3422 

3444 

3467 

349o 

2 

57 

i378 

i4oo 

1422 

1  444 

i466 

i488 

2 

58    35i2 

3535 

3557 

358o 

36o3 

3625 

I 

58 

i5o9 

i53i 

i553 

1575  i597 

:6i9 

I 

59 

3648 

367o 

3693 

37i6 

3738 

376i 

0 

59 

i64o  1662 

1  684 

1706  1728 

i749  o 

60" 

50" 

40" 

30" 

20" 

10" 

^ 

60"     50" 

40" 

30"   20"   10"  |  • 

Co-sine  of  47 

Degrees. 

2 

Co-sine  of  46  Degrees. 

i 

p.  part  |  >;  2;  3;  4; 

5"  6"  7"  8"  9" 
11  14  16  18  21 

C  1"  2"  3'-  4"  5"  6"  7"  8"  9" 
P.  Part  J  2   4   7   9  11  13  IG  18  20 

LOGARITHMIC    TANGENTS. 


B. 

Tangent  of  42  Decrees. 

A 

Tangent  of  43  Degrees. 

* 

0" 

10" 

20"  |  30"  |  40" 

50" 

a| 

0"     10" 

20" 

30" 

40" 

50" 

0 

9.954437 

448o 

4522 

4564 

4607 

4649 

59 

0 

9.969656 

9698 

974o 

9783 

9826 

0867 

59 

i 

4691 

4?34 

4776 

48i9 

486i 

4903 

58 

i 

9909 

995i 

9994 

..36 

..78 

.  120 

58 

3 

4946 

4988 

5o3o 

5073 

5n5 

5i57 

^7 

2 

9.970162 

02O5 

0247 

0289 

o33i 

o373 

57 

3 

02OO 

5242 

6284 

5327 

53b9 

i>4n 

56 

3 

o4i6 

o458 

o5oo 

0542 

o584 

0627 

56 

4 

5454 

5496 

5538 

558i 

5623 

5665 

55 

4 

0669 

0711 

o753 

0796 

o838 

0880 

55 

5 

5yo8 

5750 

5792 

5835 

5877 

59i9 

54 

5 

0922 

o964 

1007 

io49 

1091 

n33 

54 

6 

596i 

6oo4 

6o46 

6088 

6i3i 

6i73 

53 

6 

n75 

1218 

1260 

1302 

1  344 

i386 

53 

7 

62i5 

6258 

63oo 

6342 

6385 

6427 

52 

7 

1429 

i47i 

i5i3 

i555 

i597 

i64o 

52 

8 

6469 

65i2 

6554 

6596 

6639 

6681 

5i 

8 

1682 

1724 

1766 

1808 

i85i 

i893 

5i 

9 

6723 

b7bb 

6808 

b8bo 

b8g3 

b935 

5o 

9 

1935 

i977 

2019 

2062 

2IO4 

2l46 

5o 

10 

9.956977 

7020 

7062 

7104 

7i46 

7i89 

49 

10 

9.972188 

2230 

2273 

23i5 

2357 

2399 

49 

ii 

723l 

7273 

73i6 

7358 

74oo 

7443 

48 

ii 

244  1 

2484 

2526 

2568 

2610 

2652 

48 

12 

7485 

7527 

757° 

7612 

7654 

7697 

47 

12 

2695 

2737 

2779 

2821 

2863 

29o5 

47 

i3 

7739 

7781 

7823 

7866 

7908 

795o 

46 

i3 

2948 

299° 

3o32 

3o74 

3n6 

3i59 

46 

i4 

7993 

8o35 

8077 

8120 

8162 

8204 

45 

i4 

3201 

3243 

3285 

3327 

3370 

3412 

45 

i5 

8247 

8289 

833i 

8373 

84i6 

8458 

44 

i5 

3454 

3496 

3538 

358i 

3623 

3665 

44 

16 

85oo 

8543 

8585 

8627 

8670 

8712 

43 

16 

3707 

3749 

379i 

3834 

3876 

39i8 

43 

i? 

8754 

8796 

8839 

8881 

8923 

8966 

42 

17 

396o 

4OO2 

4o45 

4o87 

4i29 

4171 

4a 

18 

9008 

goSo 

9o93 

9i35 

9177 

9219 

4i 

18 

42l3 

4255 

4298 

434o 

4382 

4424 

4i 

*9 

9262 

93o4 

9346 

9389 

943i 

9473 

4o 

J9 

4466 

45o9 

455i 

4593 

4635 

4677 

4o 

20 

9.959516 

9558 

9600 

9642 

9685 

9727 

39 

20 

9.97472o 

4762 

48o4 

4846 

4888 

493o 

39 

21 

9769 

9812 

9854 

9896 

9938 

998i 

38 

21 

4973 

5oi.5 

5a57 

5o99 

5i4i 

5i83 

38 

22 

9.960023 

oo65 

0108 

oi5o 

0192 

0234 

37 

22 

5226 

5268 

53io 

5352 

5394 

5437 

37 

23 

0277 

0319 

o36i 

o4o4 

o446 

o488 

36 

23 

5479 

552i 

5563 

56o5 

5647 

569o 

36 

24 

o53o 

o573 

o6i5 

o657 

0700 

0742 

35 

24 

5732 

5774 

58i6 

5858 

59oi 

5943 

35 

25 

0784 

0826 

0869 

0911 

o953 

o996 

34 

25 

5985 

6027 

6o69 

6m 

6i54 

6i96 

34 

26 

io38 

1080 

1122 

n65 

1207 

I249 

33 

26 

6238 

6280 

6322 

6364 

6407 

6449 

33 

27 

1292 

i334 

i376 

i4i8 

i46i 

i5o3 

32 

27 

649i 

6533 

6575 

66i7 

6660 

6702 

32 

28 

1  545 

i587 

i63o 

1672 

1714 

i757 

3i 

28 

6744 

6786 

6828 

687o 

69i3 

6955 

3i 

29 

1799 

i84i 

i883 

1926 

1968 

2OIO 

3o 

29 

6997 

7o39 

7081 

7I23 

7166 

7208 

3o 

3o 

9.962052 

2095 

2137 

2179 

2222 

2264 

29 

3o 

9.97725o 

7292 

7334 

7377 

7419 

746  1 

29 

3i 

23o6 

2348 

2391 

2433 

2475 

25l7 

28 

3i 

75o3 

7545 

7587 

763o 

7672 

77i4 

28 

32 

256o 

2602 

2644 

2686 

2729 

277I 

27 

32 

7756 

7798 

784o 

7882 

7925 

7967 

27 

33 

28i3 

2856 

2898 

2940 

2982 

3o25 

26 

33 

8oo9 

8o5i 

8o93 

8i35 

8178 

8220 

26 

34 

3o67 

3109 

3i5i 

3ig4 

3236 

3278 

25 

34 

8262 

83o4 

8346 

8388 

843  1 

8473 

25 

-  35 

3520 

3363 

34o5 

3447 

3489 

3532 

24 

35 

85i5 

8557 

8599 

864i 

8684 

8726 

24 

36 

3574 

36i6 

3659 

3701 

3743 

3785 

23 

36 

8768 

8810 

8852 

8894 

8937 

8979 

23 

37 

3828 

387o 

39!2 

3954 

3997 

4o39 

22 

37 

9021 

9o63 

9io5 

9i47 

9i9o 

9232 

22 

38 

4o8i 

4123 

4i66 

4208 

425o 

4292 

21 

38 

9274 

93i6 

9358 

94oo 

9442 

9485 

21 

39 

4335 

4377 

44i9 

446  1 

45o4 

4546 

20 

39 

9527 

9569 

9611 

9653 

9695 

9738 

20 

4o 

9.964588 

463o 

4673 

47i5 

4757 

4799 

'9 

4o 

9.979780 

9822 

9864 

99o6 

9948 

999° 

r9 

4i 

4842 

4884 

4926 

4968 

5on 

5o53 

18 

4i 

9.980033 

oo75 

0117 

oi59 

0201 

0243 

18 

42 

5095 

5i37 

5i8o 

5222 

5264 

53o6 

17 

42 

0286 

o328 

o37o 

o4ia 

0454 

o496 

J7 

43 

5349 

539i 

5433 

5475 

55i8 

556o 

16 

43 

o538 

o58i 

0623 

o665 

0707 

0749 

16 

44 

56o2 

5644 

5687 

5729 

577i 

58i3 

i5 

44 

0791 

o834 

0876 

o9i8 

o96o 

IOO2 

i5 

45 

5855 

5898 

594o 

5982 

6oa4 

6067 

i4 

45 

io44 

1086 

1129 

1171 

I2l3 

1255 

i4 

46 

6109 

6i5i 

6193 

6236 

6278 

632O 

i3 

46 

1297 

i339 

1382 

i4a4 

i466 

i5o8 

i3 

4? 

6362 

64o5 

6447 

6489 

653i 

S574 

12 

47 

i55o 

l592 

1634 

1677 

1719 

1761 

12 

48 

6616 

6658 

6700 

6742 

6785 

6827 

II 

48 

i8o3 

1845 

1887 

I929 

I972 

20l4 

II 

49 

6869 

6911 

6954 

6996 

7o38 

7080 

IO 

49 

2o56 

2098 

2l4o 

2l82 

222/ 

2267 

10 

5o 

9.967123 

7i65 

7207 

7249 

729i 

7334 

9 

5o 

9.982309 

235i 

2393 

2435 

2477 

2519 

g 

5i 

7376 

74i8 

746o 

75o3 

7545 

7587 

8 

5i 

2562 

260^ 

2646 

2688 

2730 

2772 

8 

52 

7629 

7672 

7714 

7756 

7798 

784o 

7 

52 

281^ 

2857 

2899 

294l 

2983 

3o25 

7 

53 

7883 

7925 

7967 

8009 

8o52 

8o94 

6 

53 

3o67 

3io9 

3i52 

3i94 

3236 

3278 

6 

54 

8i36 

8178 

8220 

8263 

83o5 

8347 

5 

54 

3320 

3362 

34o4 

3447 

3489 

353i 

5 

55 

8389 

8432 

8474 

85i6 

8558 

8600 

4 

55 

3573 

36i5 

3657 

3699 

374a 

3784 

4 

56 

8643 

8685 

8727 

8769 

8812 

8854 

3 

56 

3826 

3868 

3910 

3952 

3994 

4o37 

3 

57 

8896 

8938 

8980 

9023 

9065 

9I07 

2 

57 

4o79 

4l2I 

4i63 

42o5 

4247 

4289 

2 

58 

9149 

9192 

9^34 

9276 

93i8 

936o 

I 

58 

4332 

4374 

44i6 

4458 

45oo 

4542 

I 

(59 

g4o3 

9445 

9487 

9529 

957i 

96i4 

O 

59 

4584 

462-7 

4669 

4711 

4753 

4795 

0 

60" 

50" 

40" 

30" 

20" 

10" 

d 

60"    |  50" 

40"   30"   20" 

10" 

. 

Co-tangent  of  47  Degrees. 

a 
i 

Co-tangent  of  46  Degrees. 

i 

p  p   «  1"  2"  3"  4"  5"  6"  7"  8"  9" 

P  Pnrt  5  l"  2"  3"  4"  5"  6"  7"  8"  9" 

}  4   8  13  17  21  25  30  34  38 

Lrl}  4   8  13  17  21  25  30  34  38 

68 


LOGARITHMIC    SINES. 


jj 

Sine  of  44  Degrees. 

a  \     Sine  of  4  5  Decrees. 

at) 

O'' 

10" 

20"  [  30"   40" 

50" 

s  |   o"   [~i<y» 

20"" 

30"" 

40" 

50'' 

0 

9.841771 

i793 

i8i5  1837  i858 

1880 

59 

0 

9.849485 

95ob 

9527 

954^ 

9569 

959o 

591 

i 

I902 

I924 

1946  1967  1989 

2OII 

58 

I 

9611 

9632 

9653 

9674 

9695 

#716 

58 

2 

2o33 

2o55 

2076  2098  2I2O 

2142  57 

2 

9738 

9759 

978o 

980; 

9822 

9843 

5y 

3 

2i63 

2iS5 

22O7  2229  225o 

2272 

56 

3 

9864 

9885 

99o6 

9927 

9948 

9969 

56 

L 

229^ 

23i6 

2337 

2359 

238i 

24oJ 

55 

4 

999° 

.  .11 

..32 

.  .53 

..74 

::95 

55 

1 

2424 

2446 

2468 

2490 

25ll 

2533 

54 

5 

9.85on6 

oi37 

oi58 

0179 

O200 

O22I 

54 

6 

2555 

2577 

2598 

2620 

2642 

2663 

53 

6 

0242 

0263 

0284 

o3o5 

o326 

0347 

53 

7 

2685 

2707 

2729 

2750 

2772 

2794 

52 

7 

o368 

o388 

o4o9 

o43o 

o45i 

0472 

52 

8 

28i5 

2837 

2859 

2880 

29O2 

2922 

5i 

8 

0493 

o5i/ 

o535 

o556 

0577 

o598 

5i 

9 

2946 

2967 

2989 

Son 

3032 

3o54 

5o 

9 

0619 

o64o 

0661 

0682 

o7o3 

0724 

5o 

10 

9.  843076 

3o97 

3119 

3i4i 

3i62 

3i84 

49 

10 

9.85o745 

0766 

0787 

0807 

0828 

o849 

49 

ii 

3206 

3227 

3249 

3271 

3292 

33i4 

48 

ii 

o87o 

o89i 

09I2 

0933 

o954 

o975 

48 

12 

3336 

3357 

3379 

34oi 

3422 

3444 

47 

12 

0996 

1017 

io38 

io58 

1070 

IIOO 

47 

i3 

3466 

3487 

35o9 

353o 

3552 

3574 

46 

i3 

II2I 

n4a 

n63 

1184 

1205 

1226 

46 

i4 

3595 

36i7 

3639 

366o 

3682 

37o3 

45 

i4 

1246 

1267 

1288 

1  309 

i33o 

i35i 

45 

i5 

3725 

3747 

3768 

379o 

38n 

3833 

44 

i5 

I372 

i393 

i4i3 

i434 

i455 

1476 

44 

16 

3855 

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3963 

43 

16 

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1601 

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17 

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1  664 

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27 

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293i 

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28 

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29 

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29 

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9.853242 

3263 

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29 

3i 

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58i2 

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28 

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3366 

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3428 

3449 

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28 

32 

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6026 

27 

32 

3490 

35ii 

3532 

3552 

3573 

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27 

33 

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6069  6090 

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6i33 

6i54 

26 

33 

36i4 

3635 

3655 

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34 

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34 

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3759 

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384i 

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35 

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35 

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48 

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48 

5465 

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49 

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573i 

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8345 

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8 

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53 

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6 

53 

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54 

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5 

54 

6201 

6221 

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55 

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4 

55 

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4 

56 

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3 

56 

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6466  6486  65o7 

6527  6547 

3 

57 

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9169 

9I90  92II 

2 

57 

6568 

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2 

58 
59 

9232 

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I 
O 

58 
59 

669o  6710  673i|675  1 
6812  6832  6853'6873 

6771  679a 
6S93  69i4 

i! 

60" 

50" 

40"  |  30" 

20"  1  10" 

. 

60"    |  50"  |  40"   30"   20"   10"   .  j 

Co-sine  of  45  Degrees. 

a 

Co-sine  of  44  Degrees.     3 

P  Part*  l"  2//  3" 

4"  5"  6"  7"  8"  9" 

(1"  2"  3"  4"  5"  6"  7"  8"  9" 

m\  2   4   G 

9  11  13  15  17  19 

'  a  £  2  4   6   8  10  12  14  17  19  i 

LOGARITHMIC      TANGENTS. 


.5 

Tangent  of  44  Degrees. 

.3 

Tangent  of  45  Degrees. 

» 

0" 

10" 

20" 

30" 

40" 

50" 

2 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.984837 

4879 

4921 

4964 

5oo6 

5o48 

59 

o 

10.000000 

OO42 

0084 

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0168 

02  I  I 

59 

i 

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5216 

5259 

53oi 

58 

I 

0253 

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0421 

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58 

2 

5343 

5385 

5427 

5469 

55n 

5553 

57 

2 

o5o5 

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o59o 

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0-716 

57 

3 

5596 

5638 

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5722 

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56 

3 

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0800 

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0884 

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o969 

56 

4 

5848 

5891 

5933 

5975 

6oi7 

6o59 

55 

4 

IOII 

io53 

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1x7.9 

1221 

55 

5 

6101 

6i43 

6i85 

6228 

6270 

63i2 

54 

5 

1263 

i3o5 

1  348 

i39o 

l432 

i474 

54 

6 

6354 

6396 

6438 

648o 

6523 

6565 

53 

6 

i5i6 

i558 

1600 

1642 

1  684 

1727 

53 

7 

66o7 

6649 

669i 

6733 

6775 

6817 

52 

7 

i769 

1811 

i853 

i895 

i937 

i979 

52 

8 

6860 

6902 

6944 

6986 

7028 

7070 

5i 

8 

2O2I 

2o63 

2106 

2148 

2I90 

2232 

5i 

9 

7II2 

7i54 

7r97 

?239 

728l 

7323 

5o 

9 

2274 

23i6 

2358 

2400 

2442 

2485 

5o 

10 

9.987365 

7407 

7449 

749i 

7534 

7576 

49 

10 

I0.002527 

2569 

2611 

2653 

2695 

2737 

49 

ii 

•7618 

7660 

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7786 

7829 

48 

ii 

2779 

2821 

2864 

29o6 

2948 

2990 

48 

12 

i3 

787i 
8i23 

79i3 
8166 

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8208 

7997 
825o 

8o39 
8292 

8081 
8334 

47 
46 

12 

i3 

3032 

3285 

3o74 
3327 

3n6 
3369 

3i58 
34n 

32OO 

3453 

3243 
3495 

47 
46 

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846o 

85o3 

8545  8587 

45 

i4 

3537 

3579 

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3664 

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8629 

867i 

87i3 

8755 

8797  884o 

44 

i5 

379o 

3832 

3874 

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3958 

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44 

16 

8882 

8924 

8966 

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9o5o 

9092 

43 

16 

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4127 

4i69 

4211 

4253 

43 

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9i34 

9i77 

9219 

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93o3 

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42 

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4295 

4337 

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4422 

4464 

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42 

18 

9387 

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95i3 

9556 

9598 

4i 

18 

4548 

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20 

9.989893 

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9977 

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39 

20 

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5222 

5264 

39 

21 

9.99oi45 

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38 

21 

53o6 

5348 

5390 

5432 

5474 

55i7 

38 

22 

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0482 

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37 

22 

5559 

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5685 

5727 

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37 

23 

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36 

23 

58n 

5854 

5896 

5938 

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6022 

36 

24 

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0988 

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1072 

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35 

24 

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6106 

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6233 

6275 

35 

25 

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1198 

1240 

1283 

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34 

25 

63i7 

6359 

64oi 

6443 

6485 

6527 

34 

26 

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i45i 

1493 

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1  5  77 

1620 

33 

26 

6569 

6612 

6654 

6696 

6738 

678o 

33 

27 

1662 

1704 

i746 

1788 

i83o 

l872 

32 

27 

6822 

6864 

6906 

6949 

699i 

7o33 

32 

28 

191,4 

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1999 

2041 

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2125 

3i 

28 

7075 

7117 

7i59 

7201 

7243 

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3i 

29 

2l67 

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225l 

2293 

2336 

2378 

3o 

29 

7328 

737o 

7412 

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7496 

7538 

3o 

3o 

9.992420 

2462 

25o4 

2546 

2588 

263o 

29 

3o 

10.007580 

7622 

7664 

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7749 

7791 

29 

3i 

2672 

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2757 

2799 

2841 

2883 

28 

3i 

7833 

7875 

7917 

7959 

8001 

8o44 

28 

32 

2925 

2967 

3009 

3o5i 

3o94 

3i36 

27 

32 

8086 

8128 

8170 

8212 

8254 

8296 

27 

33 

3i78 

322O 

3262 

33o4 

3346 

3388 

26 

33 

8338 

838o 

8423 

8465 

85o7 

8549 

26 

34 

343i 

3473 

35i5 

3557 

3599 

364i 

25 

34 

859i 

8633 

8675 

8717 

8760 

8802 

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35 

3683 

3725 

3767 

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3852 

3894 

24 

35 

8844 

8886 

8928 

9OI2 

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24 

36 

3936 

3978 

4020 

4062 

4104 

4i46 

23 

36 

9°97 

9i39 

9181 

9223 

9265 

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23 

37 

38 

4189 
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4483 

4273 

4526 

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4568 

43574399 

46io  4652 

22 
21 

37 

38 

9349 

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939i 
9644 

9433 
9686 

9476 
9?28 

95i8 
977° 

956o 
98i3 

22 
21 

39 

4694 

4736 

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4820 

4862 

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39 

9855 

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9939 

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2O 

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9-994947 

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5o3i 

5o73 

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5i57 

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4o 

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0234 

0276 

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10 

4i 

5l99 

5241 

5284 

5326 

5368 

54io 

18 

4i 

o36o 

0402 

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0629 

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18 

42 

5452 

5494 

5536 

5578 

5620 

5663 

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42 

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0781 

0823 

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43 

57o5 

5747 

5789 

583i 

5873 

59i5 

16 

43 

0866 

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o95o 

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io34 

io76 

16 

44 

5957 

5999 

6o42 

6o84 

6126 

6168 

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44 

1118 

1160 

1203 

1245 

1287 

i329 

16 

45 

6210 

6262 

6294 

6336 

6378 

6421 

i4 

45 

i37i 

i4i3 

i455 

i497 

i54o 

i582 

i4 

46 

6463 

65o5 

6547 

6589 

663i 

6673 

i3 

46 

1624 

1666 

1708 

i75o 

I792 

i834 

i3 

47 

67i5 

6757 

6800 

6842 

6884 

6926 

12 

47 

i877 

1919 

1961 

2OO3 

2o87 

12 

48 

6968 

7010 

7o52 

7o94 

7i36 

7i79 

I* 

48 

2I29 

2171 

22l4 

2256 

2298 

2340 

II 

49 

7221 

7263 

73o5 

7347 

7389 

743i 

10 

49 

2382 

2424 

2466 

25o9 

255i 

2593 

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5o 

O  •  0074-7^ 

75i5 

7558 

76oo 

7642 

7684 

9 

5o 

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2677 

2719 

276l 

28o3 

2846 

9 

5i 

7726 

7768 

78io 

7852 

7894 

7937 

8 

5i 

2888 

293o 

2972 

3oi4 

3o56 

3o98 

& 

52 

53 

7979 
823i 

8021 
8273 

8o63 
83i6 

8io5 
8358 

8i478i89 
84oo  8442 

6 

52 

53 

3i4o 
3393 

3i83 
3435 

3225 

3477 

3267 
3520 

33o9 
3562 

335i 
36o4 

6 

54 

8484 

8526 

8568 

8610 

8652 

8695 

5 

54 

3646 

3688 

3730 

3772 

38i5 

3857 

5 

55 

8737 

8779 

8821 

8863 

8905 

8947 

4 

55 

3899 

394i 

3983 

4o25 

4o67 

4io9 

4 

56 

8989 

9o3i 

9o74 

9116 

9i58  9200 

3 

56 

4i94 

4236 

4278 

4320 

4362 

3 

57 

9242 

9284 

9326 

9368 

9410  9453 

2 

$7 

44o4 

4447 

4489 

453i 

4573 

46i5 

2 

58 

9495 

9537 

9579 

9621 

96639705 

I 

58 

4657 

4699 

474i 

4784 

4826 

4868 

I 

59 

9747 

9789 

9832 

9874 

99i6  9958 

0 

59 

49io 

4952 

4994 

5o36 

5o79j5i2i 

O 

60"     50"   40" 

30"  |  20"   10" 

. 

60"      50" 

40" 

30" 

20"   10" 

. 

Co-tangent  of  45  Degrees. 

a 

Co-tangent  of  44  Degrees. 

i 

P  PnrtJ  l"  2"   3"   4"   5"  6"  7"   8"   9" 

.<  1"  2"  3"  4"  5"  6"  7"  8"  0" 

irl{  4   8  13  17  21  25  29  34  38 

in\  4   8  13  17  21  25  29  34  38 

L  O  G  A  R  I  T  IT  >T  I  C 


E  «. 


.5 

•Sine  of  46  Degrees. 

d 

Sine  of  47  Degrees. 

m 

0' 

10" 

so- 

30" 

40"   50" 

§ 

0" 

10" 

20"  |  30" 

40"  j  50" 

o  9.866934 

6954 

6975 

6995 

7016 

7o36|59 

O 

9.864i27 

4i47 

4i67 

4i86 

42O6 

4226 

69 

i 

7o56 

7076 

7°97 

7117 

7i37 

716868 

I 

4245 

4266 

4284 

43o4 

4321 

4343 

58 

2 

7178 

7198 

72I9 

7239 

7259 

7279 

57 

2 

4363 

4383 

4402 

4422 

444  1 

446i 

57 

3 

73oo 

7320 

734o 

736i 

738i 

74oi 

56 

3 

448  1 

45oo 

4620 

4539 

4559 

4579 

56 

4 

7422 

744s 

7462 

7482 

75o3 

7623 

55 

4 

4598 

46i8 

4637 

4657 

4676 

4696 

55 

5 

7543 

7563 

7584 

7604 

7624 

7645 

54 

5 

4716 

4735 

4755 

4774 

4794 

48i3 

54 

6 

7665 

7686 

7706 

7726 

7746 

7766 

53 

6 

4833 

4853 

4872 

4892 

49n 

493i 

53 

7 
8 

7786 
7908 

7807 
7928 

7827 
7948 

7847 
7968 

7867 
7989 

7888 
8009 

62 
5i 

8 

495o 
6068 

4970 
6087 

4990 
5io7 

5oo9|5o29 
6126  5i46 

5o48 
5i65 

62 
5i 

9 

8029 

8049 

8070 

8o9o 

8110 

8i3o 

5o 

9 

5i85 

6204 

6224 

5244 

5263 

5283 

5o 

10 

9.858i5i 

8171 

8i9i|82ii 

823i 

8262 

49 

10 

9.  865302 

5322 

534i 

536i 

538o 

54oo 

49 

ii 

8272 

8292 

83128332 

8353 

8373 

48 

ii 

54i9 

5439 

5458 

5478 

5497 

6617 

48 

12 

8393 

84i3 

8433 

8454 

8474 

8494 

47 

12 

5536 

5556 

5575 

5595 

56i4 

5634 

47 

i3 

85i4 

8534 

8554  8575 

8595 

86i5 

46 

i3 

5653 

5673 

5692 

57I2 

573i 

6761 

46 

i4 

8635 

8655 

8676 

8696 

8716 

8736 

45 

i4 

577o 

6790 

58o9 

6828 

5848 

5867 

45 

i5 

8756 

8776 

8796 

8817 

8837 

8857 

44 

i5 

5887 

6906 

5926 

5945 

5965 

5984 

44 

16 

8877 

8897 

8917 

8937 

8958 

8978 

43 

16 

6oo4 

6o23 

6042 

6062 

6081 

6101 

43 

i? 

8998 

9018 

9o38 

9o58 

9o78 

9o98 

42 

*7 

6120 

6i4o 

6i59 

6i79 

6i98 

621-7 

42 

18 

9119 

9i39 

9169 

9i79 

9i99 

92I9 

4i 

18 

6237 

6266 

6276 

6295 

63i5 

6334 

4i 

r9 

9239 

9259 

9279 

93oo 

9320 

934o 

4o 

19 

6353 

6373 

6392 

6412 

643i 

645o 

4o 

20 

9.869360 

938o 

9400  9420 

944o 

946o 

39 

20 

9.86647o 

6489 

65o9 

6528 

6547 

6567 

39 

21 

9480 

95oi 

9621 

9641 

956i 

9681 

38 

21 

6586 

6606 

6626 

6644 

6664 

6683 

38 

22 

9601 

962I 

964i 

966i 

968i 

97oi 

37 

22 

67o3 

6722 

674i 

676i 

678o 

6800 

37 

23 

9721 

9?4i 

9761 

978i 

9802 

9822 

36 

23 

6819 

6838 

6858 

6877 

6896 

69i6 

36 

24 

9842 

9862 

9882  9902 

9922 

9942 

35 

24 

6935 

6954 

6974 

6993 

7oi3 

7o32 

35 

25 

9962 

9982 

.  .  .2 

.  .22 

..42 

..62 

34 

25 

7o5i 

7o7i 

7o9o 

7io9 

7i29 

7i48 

34 

26 

9.860082 

OIO2 

OI22  Ol42 

Ol62 

0182 

33 

26 

7i67 

7187 

•7206 

7225 

7245 

7264 

33 

27 

O2O2 

O222 

0242  0262 

0282 

0302 

32 

27 

7283 

73o3 

7322 

734i 

736i 

738o 

32 

28 

O322 

0342 

o362o382 

0402 

0422 

3i 

28 

7399 

74i9 

7438 

7457 

7476 

7496 

3i 

29 

0442 

0462 

0482  0602 

O522 

0642 

3o 

29 

75!5 

7534 

7554 

7573 

7592 

76l2 

3o 

3o 

9.860662 

0582 

0602  0622 

0642 

0662 

29 

3o 

9.  867631 

7660 

7669 

7689 

7708 

7727 

29 

3i 

0682 

0702 

0722  0742 

0762 

0782 

28 

3i 

7747 

7766 

7785 

78o4 

7824 

7843 

28 

32 

0802 

0822 

08420862 

0882 

0902 

27 

32 

7862 

7882 

79oi 

7920 

7939 

7959 

27 

33 

0922 

o94i 

0961 

o98i 

1001 

1021 

26 

33 

7978 

7997 

8016 

8o36 

8066 

8o74 

26 

34 

io4i 

1  06  1 

1081 

IIOI  II2I 

I  l4l 

25 

34 

8o93 

8n3 

8i32 

8i5i 

8170 

8i9o 

26 

35 

1161 

1181 

I2OI 

1221 

1240 

1260 

24 

35 

8209 

8228 

8247 

8267 

8286 

83o5 

24 

36 

1280 

i3oo 

1320 

i34o 

i36o 

!38o 

23 

36 

8324 

8343 

8363 

8382 

84oi 

8420 

23 

37 

i4oo 

i4so 

i439 

i459 

1479 

1499 

22 

37 

844o 

845o' 

8478 

8497 

85i6 

8536 

22 

38 

1619 

i539 

1669 

i579 

i599 

1618 

21 

38 

8555 

8574 

8593 

8612 

8632 

8661 

21 

39 

i638 

1668 

1678 

i698 

1718 

i738 

2O 

39 

8670 

8689 

8708 

8?28 

8747 

8766 

2O 

4o 

9.86:768 

1777 

1797 

1817 

1837 

i857 

'9 

4o 

9.868786 

88o4 

8823 

8843 

8862 

8881 

:9 

4  1 

1877 

i897 

1916 

I936 

1966 

i976 

18 

4i 

8900 

89i9 

8939 

8968 

8977 

8996 

1  8 

43 

1996 

2016 

2o35  2066 

2076 

2095 

r7 

42 

9016 

9o34 

9o53 

9°73 

9O92 

9111 

i? 

43 

2Il5 

2!35 

2i54 

2174 

2194 

22l4 

16 

43 

9i3o 

9i49 

9I68 

9i88 

9207 

9226 

16 

44 

2234 

2254 

2273  2293 

23i3 

2333 

i5 

44 

9245 

9264 

9283 

9302 

932I 

934i 

16 

45 

2353 

2372 

2302 

2412 

2432 

2462 

i4 

45 

936o 

9379 

9398 

94i7 

9436 

9455 

i4 

46 

2471 

s49i 

25ll 

253i 

2661 

257O 

i3 

46 

9474 

9494 

95i3 

9532 

955i 

957o 

i3 

4? 

•2690 

2610 

263o 

2660 

2669 

2689 

12 

47 

9689 

96o8 

9627 

9646 

9665 

9685 

12 

48 

2709 

2729 

2  748 

2768 

2788 

2808 

II 

18 

9704 

9723 

9742 

976i 

9780 

9799 

II 

49 

2827 

2847 

2867 

2887 

29o6 

2926 

IO 

49 

9818 

9837 

9866 

9875 

9894 

9914 

10 

5o 

9.862946 

2966 

2985 

3oo5 

3o25 

3o45 

9 

5o 

9.869933 

9952 

997r 

999° 

...9 

..28 

9 

5i 

3o64 

3o84 

3io4 

3124 

3i43 

3i63 

8 

5i 

9.870047 

0066 

0086 

oio4 

0123 

0142 

8 

52 

3i83 

32o3 

3222 

3242 

3262 

3281 

7 

62 

0161 

0180 

oi99 

02x8 

0233 

0257 

7 

53 

33oi 

332i 

334i 

336o 

338o 

34oo 

6 

53 

0276 

O295 

o3i4 

o333 

o352 

o37i 

6 

54 

34i9 

3439 

3459 

3478 

3498 

35!8 

5 

54 

0390 

o4o9 

0428 

0447 

o466 

o485 

5 

55 

3538 

3557 

3577 

3597 

36i6 

3636 

4 

55 

o5o4 

0623 

0642  0661  0680 

0699 

4 

M 

s? 

3656 
3774 

3675 
3793 

3695 
38i3 

3716 
3833 

3734 
3852 

3754 
3872 

3. 

2 

56 

57 

06180637 
0732  0761 

0666  o675  0694 
o77o  o789  0808 

o7i3  3 

0827   2 

58 

3892 

39n 

393i 

395i 

397o 

399o 

I 

58 

o846  0866 

o884  0903)0932  o94  1  i 

Us. 

4oio 

4o29 

4o49 

4o69 

4o88 

4io8 

0 

59 

0960)0979  o998|ioi7  io36  io54  ° 

60"    j  50" 

40" 

30"  |  20" 

10" 

d 

60"     50"  |  40"   30"   20"   10"   _j 

Co-sine  of  43 

Degrees. 

§ 

Co-sine  of  42  Degrees. 

(  I//   o"   Q"   4" 

**?.{•  "4   6   8 

5"  6"  7"  8"  9" 
10  12  14  16  18 

<  1"  2"  3"  4"  5"  G"  7"  8"  ti"  \ 
^•lart}  o   4   G   8  10  12  14  15  17 

LOGARITHMIC    TANGENTS. 


1 

[    Tangent  of  46  Degrees. 

* 

Tangent  of  47  Degrees. 

i 

0" 

10" 

20" 

30" 

40" 

50" 

3 

0" 

10"  i  20" 

30"   40" 

50" 

o 

io.oi5i63 

52o5 

5247 

5289 

533i 

5373 

59 

0 

io.o3o344 

o386 

0429 

o47i 

o5i3 

o555 

59 

I 

54i6 

5458 

55oo 

5542 

5584 

5626 

58 

I 

o597 

o64o 

0682 

0724 

o766 

0808 

58 

2 

5668 

57n 

5753 

5795 

5837 

5879 

57 

2 

o85i 

o893 

o935 

°977 

1020 

1062 

57 

3 

5921 

5963 

6006 

6048 

6o9o 

6i32 

56 

3 

no4 

n46 

1188 

I23l 

I273 

i3i5 

56 

4 

6i74 

6216 

6258 

63oi 

6343 

6385 

55 

4 

i357 

i4oo 

i442 

i484 

i526 

i568 

55 

5 

6427 

6469 

65n 

6553 

6596 

6638 

54 

5 

1611 

i653 

i695 

:737 

1780 

1822 

54 

6 

6680 

6722 

6764 

6806 

6848 

689i 

53 

6 

1864 

i9o6 

i948 

I99I 

2033 

2O75 

53 

7 

6933 

6975 

7017 

7o59 

7IOI 

7i43 

52 

7 

21  I7 

2160 

22O2 

2344 

2286 

2328 

52 

8 

7186 

7228 

7270 

73l2 

7354 

7396 

5i 

8 

2371 

24i3 

2455 

2497 

254o 

2582 

5i 

9 

7438 

748  1 

7523 

7.565 

76o7 

7649 

5o 

9 

2624 

2666 

2709 

275l 

2793 

2835 

5o 

10 

10.017691 

7733 

7776 

78i8 

7860 

7902 

49 

10 

IO.O32877 

2920 

2962 

3oo4 

3o46 

3o89 

49 

ii 

7944 

7986 

8028 

8o7i 

8n3 

8i55 

48 

i  i 

3i3i 

3i73 

32i5 

3258 

33oo 

3342 

48 

12 

8I97 

8239 

8281 

8323 

8366 

84o8 

47 

12 

3384 

3426 

3469 

35n 

3553 

3595 

47 

i3 

845o 

8492 

8534 

8576 

8618 

8661 

46 

i3 

3638 

368o 

3722 

3764 

38o7 

3849 

46 

i4 

87o3 

8745 

8787 

8829 

8871 

89i4 

45 

i4 

389i 

3933 

3976 

4oi8 

4o6o 

4l02 

45 

i5 

8956 

8998 

9o4o 

9o82 

9I24 

9i66 

44 

i5 

4i45 

4i87 

4229 

427I 

43i3 

4356 

44 

16 

9209 

925i 

9293 

9335 

9377 

94i9 

43 

16 

4398 

444o 

4482 

4525 

4567 

46o9 

43 

i? 

9462 

95o4 

9546 

9588 

963o 

9672 

42 

«7 

465  1 

4694 

4736 

4778 

4820 

4863 

42 

18 

97i4 

9757 

9799 

984i 

9883 

9925 

4i 

18 

49o5 

4947 

4989 

5o32 

5o74 

5n6 

4i 

J9 

9967 

.  .10 

.  .62 

..94 

.136 

.178 

4o 

19 

5i58 

52OI 

5243 

5285 

5327 

537o 

4o 

20 

IO.O20220 

0262 

o3o5 

o347 

o389 

o43i 

39 

20 

io.o354i2 

5454 

5496- 

5539 

558i 

5623 

39 

21 

0473 

o5i5 

o558 

0600 

0642 

o684 

38 

21 

5665 

57o8 

575o 

5792 

5834 

5877 

38 

22 

0726 

0768 

0810 

o853 

o895 

°937 

37 

22 

59i9 

596i 

6oo3 

6o46 

6088 

6i3o 

37 

23 

0979 

1021 

io63 

1106 

n48 

II90 

36 

23 

6172 

62i5 

6257 

6299 

634i 

6384 

36 

24 

1232 

1274 

i3i6 

i359 

i4oi 

i443 

35 

24 

6426 

6468 

65ii 

6553 

6595 

6637 

35 

25 

i485 

1527 

i569 

1612 

i654 

i696 

34 

25 

6680 

6722 

6764 

6806 

6849 

689i 

34 

26 

i738 

1780 

1822 

i865 

1007 

i949 

33 

26 

6933 

6975 

7oi8 

7o6o 

7IO2 

7i44 

33 

27 

1991 

2033 

2075 

2118 

2160 

2202 

32 

27 

7187 

7229 

727I 

73i4 

7356 

7398 

32 

28 

2244 

2286 

2328 

237o 

24i3 

2455 

3i 

28 

744o 

7483 

7525 

7567 

76o9 

7652 

3i 

29 

2497 

2539 

258i 

2623 

2666 

2708 

3o 

29 

7694 

7736 

7778 

782I 

7863 

79o5 

3o 

3o 

10.022750 

2792 

2834 

2877 

29I9 

296i 

29 

3o 

io.o37948 

799° 

8o32 

8o74 

8117 

8i59 

29 

3i 

3oo3 

3o45 

3087 

3i3o 

3172 

3214 

28 

3i 

8201 

8243 

8286 

8328 

837o 

84i3 

28 

32 

3256 

3298 

334o 

3383 

3425 

3467 

27 

32 

8455 

8497 

8539 

8582 

8624 

8666 

27 

33 

3509 

355i 

3593 

3636 

3678 

3720 

26 

33 

8708 

875i 

8793 

8835 

8878 

8920 

26 

34 

3762 

38o4 

3846 

3889 

3o3i 

3973 

25 

34 

8962 

9oo4 

9o47 

9o89 

9i3i 

9i74 

25 

35 

4oi5 

4o57 

4o99 

4:42 

4i84 

4226 

24 

35 

92l6 

9258 

93oo 

9343 

9385 

9427 

24 

36 

4268 

43io 

4353 

4395 

4437 

4479 

23 

36 

947o 

95l2 

9554 

9596 

9639 

968i 

23 

37 

452i 

4563 

46o6 

4648 

469o 

4?32 

22 

37 

9723 

9766 

98o8 

985o 

9892 

9935 

22 

38 

4774 

48i7 

4859 

49oi 

4943 

4985 

21 

38 

9977 

..i9 

..62 

.  io4 

.i46 

.188 

21 

39 

5027 

5o7o 

5lI2 

5i54 

5i96 

5238 

20 

39 

io.o4o23i 

0273 

o3i5 

o358 

o4oo 

o442 

2O 

4o 

10.025280 

5323 

5365 

54o7 

5449 

5491 

I  9 

4o 

io.o4o484 

0527 

o569 

0611 

o654 

o696 

I9 

4i 

5534 

5576 

56i8 

566o 

57O2 

5745 

18 

4i 

0738 

0781 

0823 

o865 

o9o7 

o95o 

18 

42 

5787 

5829 

58yi 

59i3 

5955 

5998 

i? 

42 

0992 

io34 

io77 

in9 

1161 

1204 

J7 

43 

6o4o 

6082 

6124 

6166 

62O9 

625i 

16 

43 

1246 

1288 

i33o 

i373 

i4i5 

i457 

16 

44 

6293 

6335 

6377 

64i9 

6462 

65o4 

i5 

44 

i5oo 

1  542 

1  584 

l627 

i669 

1711 

1  5 

45 

6546 

6588 

663o 

6673 

67i5 

6757 

i4 

45 

1753 

i796 

i833 

1880 

I923 

I965 

i4 

46 

6799 

684  1 

6884 

6926 

6968 

7010 

i3 

46 

2007 

2o5o 

2O92 

2134 

2177 

22I9 

i3 

47 

7062 

7°95 

7i37 

7179 

7221 

7263 

12 

47 

2261 

23o3 

2346 

2388 

243o 

2473 

12 

48 

73o5 

7348 

739o 

7432 

7474 

75i6 

I  I 

48 

25i5 

2557 

26OO 

2642 

2684 

2727 

II 

49 

7559 

7601 

7643 

7685 

7727 

7770 

10 

49 

2769 

2811 

2854 

2896 

2938 

298o 

10 

So 

10.027812 

7854 

7896 

7938 

7981 

8o23 

9 

5o 

io.o43o23 

3o65 

3io7 

3i5o 

3i92 

3234 

9 

5.i 

8o65 

8107 

8r49 

8i92 

8234 

8276 

8 

5i 

3277 

33i9 

336i 

34o4 

3446 

3488 

8 

52 

83i8 

836o 

84o3 

8445 

8487 

8529 

7 

52 

353i 

3573 

36i5 

3658 

37oo 

3742 

7 

53 

857i 

8614 

8656 

8698 

874o 

8782 

6 

53 

3785 

3827 

3869 

39I2 

3954 

3996 

6 

54 

8825 

8867 

89o9 

895i 

8993 

9o36 

5 

54 

4o39 

4o8i 

4123 

4x65 

4208 

425o 

5 

55 

9078 

9I20 

9162 

9204 

9247 

9289 

4 

55 

4292 

4335 

4377 

44i9 

4462 

45o4 

4 

55 

933! 

9373 

94i6 

9458 

95oo 

9542 

3 

56 

4546 

4589 

463i 

4673 

47i6 

4758 

3 

57 

9584 

9627 

9669 

9711 

9753 

9795 

2 

57 

48oo 

4843 

4885 

4927 

497o 

5OI2 

2 

58 

9838 

988o 

9922 

9964 

...6 

..49 

I 

58 

5o54 

5o97 

5i39 

5i8i 

5224 

5266 

I 

59 

io.o3oo9i 

oi33 

oi75 

02I7 

0260 

0302 

O 

59 

53o9 

535i 

5393 

5436 

5478 

5520 

0 

60" 

50" 

40" 

30" 

20" 

10" 

a 

60"     50"  j  40" 

30" 

20" 

10" 

c; 

Co-tangent  of  43  Degrees. 

a 

Co-tangent  of  42  Degrees. 

i 

p  ,,   $  I"  2"  3"  4"  5"  6"  7"  8"  9" 
<  -1   8  13  17  21  25  30  34  38 

P  PirtJ  r/  2"  3"  4"  5"  6//  7"  8"  9" 
\  4   8  13  17  21  25  30  34  38 

72 


LOGARITHMIC    SINES. 


d 

Sine  of  48  Degrees. 

.8 

Sine  of  49  Degrees. 

i 

0" 

10" 

20"   30"   40' 

50' 

5 

O' 

10"   20"   30" 

40' 

50" 

0 

9.871073 

1092 

1  1  ii  n3o  1  149 

1168 

59 

0 

9.  877780 

7798 

78l6 

7835 

7853 

7871 

59 

i 

1187 

1206 

1225  1244  1263 

1282 

58 

I 

7890 

79o8 

7926 

7945 

7963 

798i 

58 

2 

i3oi 

l320 

i33g  i358  i377 

i395 

57 

2 

7999 

8018 

8o36 

8o54 

8072 

8091 

57 

3 

i4i4 

i433 

i452  i47i  i49o 

iSog 

56 

3 

8109 

8127 

8i46 

8i64 

8182 

8200 

56 

4 

i528 

i547 

1566^585 

i6o4 

1622 

55 

4 

8219 

8237 

8255 

8273 

8292 

83io 

55 

5 
6 

i64i 
1755 

1660 
i774 

1679^1698  1717 
1793  i8nli83o 

1736 
1849 

54 
53 

5 
6 

8328 
8438 

8346 
8456 

8365 
8474 

8383 
8492 

84oi 
85n 

8419 
8529 

54 
53 

'  7 

1868 

1887  1906  1925  1944 

1962 

52 

7 

8547 

8565 

8583 

8602 

8620 

8638 

52 

8 

1981 

2OOO 

2019  2o38  2057 

2076 

5i 

8 

8656 

8675 

8693 

8711 

8729 

8747 

5i 

9 

2095 

2Il3  2l32  2l5l 

2170 

2i89 

5o 

9 

8766 

8784 

8802 

8820 

8838 

8857 

5o 

10 

9.872208 

2226  2245  2264  2283 

23O2 

49 

10 

9.878875 

8893 

8911 

8929 

8948 

8966 

49 

ii 

2321 

2340235828772396 

24i5 

48 

ii 

8984 

9002 

9020 

9o39 

9o57 

9075 

48 

12 

2434 

2452  2471 

2490'2509 

2528 

47 

12 

9o93 

9111 

9129 

9148 

9166 

9184 

47 

i3 

2547 

2565  258426032622 

264l 

46 

i3 

9202 

9220 

9238 

9257 

92?5 

9293 

46 

i4 

2659 

2678  26972716 

2735 

2753 

45 

i4 

93n 

9329 

9347 

9365 

9384 

9402 

45 

i5 

2772 

2791 

2810  2829 

2847 

2866 

44 

i5 

9420 

9438 

9456 

9474 

9492 

95n 

44 

16 

2885 

2904 

2923|294l 

2960 

2979 

43 

16 

9529 

9547 

9565 

9583 

9601 

9619 

43 

18 

2998 
3no 

3oi63o35  3o54 
3i293i48|3i66 

3o73 
3i85 

3o9i 
3204 

42 

4i 

18 

9637 
9746 

9656 
9?64 

9674 

9782 

9692 
98oo 

97io 
9819 

9728 

9837 

42 

4i 

19 

3223 

324l 

3260  3279 

3298 

33i64o 

'9 

9855 

9873 

989i 

99°9 

9927 

9945 

4o 

20 

9.873335 

3354;3373 

339i 

34io 

3429  39 

20 

9.879963 

9981 

..18 

..36 

..54 

39 

21 

3448 

34663485 

35o4 

3522 

354i 

38 

21 

9.88oo72  oo9o 

0108 

0126 

oi44 

0162 

38 

22 

356o 

3579!3597 

36i6 

3635 

3653 

37 

22 

0180 

0198 

0216 

0234 

0253 

027I 

37 

23 

3672 

369i 

3710 

3728 

3747 

3766 

36 

23 

O289 

0307 

o325 

o343 

o36i 

o379 

36 

24 

3784 

38o3 

3822  384o 

3859 

3878 

35 

24 

o397 

o4i5 

o433 

o45i 

0469 

o487 

35 

25 

3896 

39i5 

3934l3953 

397i 

399o 

34 

25 

o5o5 

o523 

o54i 

o559 

o577 

o595 

34 

26 

4009 

4o27  4o46'4o65 

4o83 

4l02 

33 

26 

06  1  3 

o63i 

o649 

0667 

0686 

0704  33 

27 

4l2I 

4i58 

4177 

4i95 

4214 

32 

27 

0722 

0740 

o758 

0776 

o794 

0812  32 

28 

4232 

42  5  1 

4270 

4288 

43o7 

4326 

3i 

28 

o83o 

0848 

0866 

o884 

0902 

0920 

3i  1 

29 

4344 

4363^4382 

44oo 

44i9 

4438 

3o 

29 

o938 

0956 

o974 

O992 

1010 

1028 

3o 

3o 

9.874456 

4475 

4493 

45i2 

453i 

4549 

2O 

3o 

9.881046 

io63 

1081 

io99 

1117 

n35 

2Q 

3i 

4568 

458646o5 

4624 

4642 

466i 

2o 

3i 

n53 

1171 

n89 

I207 

1225 

1243 

2$ 

32 

468o 

46984717 

4735 

4754 

4773 

27 

32 

1261 

1279 

i297 

i3i5 

i333 

i35i 

27 

33 

4791 

48104828 

4847 

48664884 

26 

33 

1  369 

i387 

i4o5 

i423 

i44i 

i459 

25 

34 

4903 

492I 

4g4o 

4958 

4977 

4996 

25 

34 

i477 

1495 

l5l2 

i53o 

1  548 

i566 

25 

35 

5oi4 

5o33  5o5i 

5070 

5o88 

5107 

24 

35 

1  584 

1602 

1620 

i638 

i656 

1674 

24 

36 

5i26 

5i44 

5i63 

5i8i 

52OO 

5218 

23 

36 

1692 

1710 

1-728 

i746 

i763 

1781 

23 

37 

5237 

5255 

5274 

5293 

53n 

533o 

22 

37 

i799 

1817 

i835 

i853 

1871 

1889 

22 

38 

5348 

53675385 

54o4 

5422 

544  1 

21 

38 

1907 

1925 

I942 

i96o 

i978 

1996 

21 

39 

5459 

5478 

5496  55i5 

5534 

5552 

2O 

39 

20l4 

2032 

2o5o 

2068 

2086 

2103 

2O 

4o 

9.87557i 

5589  56o8 

5626 

5645  5663 

19 

4o 

9.882121 

2139 

2l57 

2I75 

2I93 

2211 

I9 

4i 

5682 

57oo 

57J9 

5737 

57565774 

18 

4i 

2229 

2246 

2264 

2282 

23oo 

23i8 

18 

42 

5793 

58n 

583o 

5848 

5867 

5885 

17 

42 

2336 

2354 

237I 

2389 

240-7 

2425 

17 

43 

59o4 

5922 

594i 

5959 

59-78  5996 

16 

43 

2443 

2461 

2479 

2496 

25i4 

2532 

16 

44 
45 

6014 
6125 

6o33 

6i44 

6o5i 
6162 

6o7o 
6181 

6088  6io7 

i5 

:4 

44 

45 

255o 
2657 

2568 
2675 

2586 
2692 

26o3 

27IO 

2621 

2-728 

2639 
2746 

i5 

46 

6236 

6255 

6273  629i 

63io6328 

i3 

46 

2764 

2782 

2799 

281-7 

2835 

2853 

i3 

47 

6347 

6365 

63846402 

6421  6439 

12 

47 

28-71 

2888 

2906 

2924 

2942 

2960 

12 

48 

6457 

6476 

64Q465i3 

653i  655o 

I  I 

48 

2977 

2995 

3oi3 

3o3i 

3o49 

3o66 

II 

49 

6568 

6586(66o5 

6623 

6642  6660 

10 

49 

3o84 

3l02 

3l20 

3i37 

3i55 

3i73 

IO 

5o 

9.  876678 

669-7 

67i5 

6734 

6752  6770 

9 

5o 

9.883191 

3209 

3226 

3244 

3262 

3280 

9 

5i 

6789 

68o7 

6826 

6844 

68626881 

8 

5i 

3297 

33i5 

3333 

335i 

3368 

3386 

8 

52 

53 

6899 
•7010 

6918 

7028 

6936 
7046 

6954 
7o65 

6973^991 
7o83|7ioi 

6 

52 

53 

34o4 
35io 

3422 
3528 

3439 
3546 

3457 
3564 

3475 
358i 

3493 
3599 

7 
6 

54 

7I2O 

7i38 

7l57 

7175 

7193-7212 

5 

54 

36i7 

3635 

3652 

367o 

3688 

37o5 

5 

55 

723o 

7248 

7267 

7280 

73o3  7322 

4 

55 

372.? 

374i 

3759 

3776 

3794 

38  1  2 

4 

56 

734o 

7358 

7?,95 

74i3  743z 

3 

56 

3829 

3847 

3865 

3883 

3900 

3918 

3 

57 

745o 

7468 

7487 

75o5 

7523  7542 

2 

57 

3936 

3953 

397i  3989 

4oo6 

4024 

2 

58 
59 

756o 
767o 

7578  7597 
-7688  77o7 

76i5 
7725 

76337653 

7743:7762 

I 

o 

58 
59 

4042 

4i48 

40604077  4095  4n34i3o 
4i66  418342014219  4236 

I 
0 

60" 

50" 

40"   30" 

20"  ;  10" 

g 

6D" 

50"   40"   30"   20"   10"   j 

Co-sine  of  41  Degrees. 

Co-sine  of  40  Degrees. 

* 

(  1"  2"  3' 

'  4"  5"  6"  7"  8"  9"  II  p  p  t$  1"  2"  3"  4"  5"  6"   "  8"  9" 

P'Par(  2   4   6 

7   9   11  13  15  17  !!  P'Part}  24   5   7   9  11  13  14  1*> 

LOGARITHMIC    TANGENTS. 


73 


Jj 

Tangent  of  48  Degree's. 

a 

IT? 

Tangent  of  49  Degrees. 

0"      10"  |  20"  |  30" 

40" 

50" 

2 

0" 

10* 

20" 

30" 

40" 

50" 

o 

io.o45563 

56o5 

5647 

569o 

5732 

5774 

r 

0 

io.o6o837 

o879 

0922 

o965 

IOO7 

io5o 

59 

I 

58i7 

5859 

59oi 

5944 

5986 

6028 

58 

i 

1092 

n35 

1177 

1220 

1262 

i3o5 

58 

2 

6o7i 

6n3 

6i55 

6i98 

6240 

6282 

57 

2 

i347 

i39o 

1432 

i475 

i5i7 

i56o 

57 

3 

6325 

6367 

64o9 

6452 

6494 

6537 

56 

3 

1602 

1  645 

1688 

i73o 

1773 

i8i5 

56 

4 

6579 

6621 

6664 

6706 

6748 

679i 

55 

4 

i858 

I900 

i943 

i985 

207O 

55 

5 

6833 

6875 

69i8 

696o 

7002 

7o45 

54 

5 

2Il3 

2i55 

2I98 

224l 

2283 

2326 

54 

6 

7087 

7i3o 

7I72 

7214 

7257 

7299 

53 

6 

2368 

2411 

2453 

2496 

2538 

258i 

53 

7 

7384 

7426 

7468 

7553 

52 

7 

2623 

2666 

2709 

275l 

2794 

2.836 

52 

8 

7595 

7638 

768o 

7723 

7765 

78o7 

5i 

8 

2879 

292I 

2964 

3oo6 

3o49 

3o92 

5i 

9 

785o 

7892 

7934 

7977 

8oi9 

8062 

5o 

9 

3x34 

3i77 

32I9 

3262 

33o4 

3347 

5o 

10 

i  o  .  o48  i  o4 

8i46 

8i89 

823i 

8273 

83i6 

49 

10 

io.o63389 

3432 

3475 

35i7 

356o 

36o2 

49 

ii 

8358 

84oo 

8443 

8485 

8528 

857o 

48 

ii 

3645 

3687 

373o 

3773 

38i5 

3858 

48 

12 

i3 

8612 

8867 

8655 
89o9 

8697 
895i 

8739 
8994 

8782 
9o36 

8824 
9°79 

47 
46 

12 

1  3 

39oo 
4i56 

3943 
4i98 

3985 

4028 
4283 

4070  4n3 
43264368 

47 
46 

i4 

9I2I 

9i63 

9206 

9248 

9290 

9333 

45 

i4 

44n 

4454 

4496 

4539 

458i 

4624 

45 

i5 

9375 

94i8 

9460 

95o2 

9545 

9587 

44 

i5 

4667 

4709 

47.52 

4794 

4837 

4879 

44 

16 

9629 

9672 

97i4 

9757 

9799 

984i 

43 

16 

4922 

4965 

5007 

5o5o 

5o92 

5i35 

43 

T7 

9884 

9926 

9969 

.  .  ii 

..53 

..96 

42 

17 

5i78 

5220 

5263 

53o5 

5348 

539o 

42 

18 

io.o5oi38 

0181 

0223 

0265 

o3o8 

o35o 

4i 

18 

5433 

5476 

55i8 

556i 

56o3 

5646 

4i 

I9 

o392 

o435 

0477 

O52O 

o562 

o6o4 

4o 

i9 

5689 

5731 

5774 

58i6 

5S59 

59O2 

4o 

20 

io.o5o647 

o689 

0732 

0774 

0816 

o859 

39 

20 

io.o65944 

5987 

6o29 

6o72 

6n5 

6i57 

39 

21 

o9oi 

o944 

o986 

1028 

1071 

in3 

38 

21 

6200 

6242 

6285 

6328 

637o 

64i3 

38 

22 

n56 

1198 

1240 

1283 

i325 

i368 

37 

22 

6455 

6498 

654i 

6583 

6626 

6668 

37 

23 

i4io 

1452 

i495 

i537 

i58o 

1622 

36 

23 

67n 

6754 

6796 

6839 

6881 

6924 

36 

24 

i665 

1707 

1749 

I792 

i834 

i877 

35 

24 

6967 

7oo9 

7o52 

7o94 

7i37 

7i8o 

35 

25 

i9i9 

1961 

2004 

2046 

2o89 

2l3l 

34 

25 

•7222 

7265 

73o8 

735o 

7393 

7435 

34 

26 

2I73 

2216 

2258 

2301 

2343 

2386 

33 

26 

7478 

752I 

7563 

7606 

7649 

769i 

33 

27 

2470 

25i3 

2555 

2598 

2640 

32 

27 

7734 

7776 

7819 

7862 

79°4 

7947 

32 

28 

2682 

2725 

2767 

2810 

2852 

2895 

3i 

28 

799° 

8o32 

8o75 

8117 

8160 

8203 

3i 

29 

2937 

2979 

3022 

3o64 

3107 

3i49 

3o 

29 

8245 

8288 

833i 

8373 

84i6 

8458 

3o 

3o 

i<  ;53i92 

3234 

3276 

33i9 

336i 

34o4 

29 

3o 

io.o685oi 

8544 

8586 

8629 

8672 

87i4 

29 

3i 

3446 

3489 

353i 

3573 

36i6 

3658 

28 

3i 

8757 

8800 

8842 

8885 

8927 

897o 

28 

33 

37oi 

3743 

3786 

3828 

387o 

39i3 

27 

32 

9013 

9o55 

9o98 

9i4i 

9183(9226 

27 

3SJ     3955 

3998 

4o4o 

4o83 

4i25 

4i68 

26 

33 

9269 

93n 

9354 

9397 

9439  9482 

26 

34     4210 

4252 

4295 

4337 

438o 

4422 

25 

34 

9525 

9567 

96io 

9652 

96959738 

25 

:5;    4465 

45o7 

4549 

4592 

4634 

4677 

24 

35 

978o 

9823 

9866 

99o8 

995ij9994 

24 

36 

4719 

4762 

48o4 

4847 

4889 

493i 

23 

36 

10.070036 

oo79 

0122 

0164 

02O7i025o 

23 

37 

4974 

5oi6 

5o59 

5ioi 

5i44 

5i86 

22 

3? 

O292 

o335 

o378 

0420 

o463 

o5o6 

22 

38 

527I 

53i4 

5356 

5398 

544i 

21 

38 

o548 

o59i 

o634 

o676 

o7i9 

O762 

21 

39 

5483 

5526 

5568 

56n 

5653 

5696 

20 

39 

0804 

o847 

o89o 

0932 

°975 

1018 

2O 

4o 

io.o55738 

578i 

5823 

5865 

59o8 

595o 

19 

4o 

10.071060 

no3 

n46 

1188 

I23l 

I274 

'9 

4i 

5993 

6o35 

6078 

6120 

6i63 

6205 

18 

4i 

i3i6 

i359 

I4O2 

i445 

i487 

i53o 

18 

42 

6248 

629o 

6333 

6375 

64i7 

646o 

I7 

42 

i573 

i6i5 

i658 

I7OI 

1743 

i786 

17 

43 

65o2 

6545 

6587 

663o 

6672 

67i5 

16 

43 

l829 

1871 

i9i4 

i957 

i999 

2O42 

16 

44 

6757 

6800 

6842 

6885 

6927 

697o 

i5 

44 

2085 

2128 

2I7O 

22l3 

2256 

2298 

i5 

45 

7012 

7o55 

7°97 

7i39 

7182 

7224 

i4 

45 

234l 

2384 

2426 

2469 

25l2 

2554 

i4 

46 

7267 

73o9 

7352 

7394 

7437 

7479 

i3 

46 

2597 

2640 

2683 

2725 

2768 

2811 

i3 

47 

7522 

7564 

7607 

7649 

7692 

7734 

12 

47 

2853 

2806 

2939 

298i 

3o24l3o67 

12 

48 

7777 

78i9 

7862 

79°4 

7947 

7989 

II 

48 

3no 

3i52 

3i95 

3238 

3280  3323 

II 

49 

8o32 

8o74 

8117 

8i59 

8202 

8244 

IO 

49 

3366 

34o9 

345  1 

3494 

3537 

3579 

IO 

5o 

10.068287 

8329 

8372 

84i4 

8456 

8499 

9 

5o 

10.073622 

3665 

37o8 

375o 

3793 

3836 

9 

5i 

854i 

8584 

8626 

8669 

8711 

8754 

8 

5i 

3878 

392I 

3964 

4oo7 

4o49 

4092 

8 

52 

8796 

8839 

8881 

8924 

8966 

9oo9 

7 

52 

4i35 

4178 

4220 

4263 

43o6 

4348 

7 

53 

9o5i 

9o94 

9i36 

9i79 

922I 

9264 

6 

53 

439: 

4434 

4477 

45i9 

4562 

46o5 

6 

54 

93o6 

9349 

939i 

9434 

9476 

95l9 

5 

54 

4648 

4S9o 

4733 

4776 

48i9 

486i 

5 

55 

956i 

96o4 

9646 

9689 

9774 

4 

55 

49o4 

4947 

4989 

5o32 

5o75 

5nd 

4 

56 

98i7 

9859 

9902 

9944 

9987 

..29 

3 

56 

5i6o 

52o3 

5246 

5289 

533i 

5374 

3 

57 

10.060072 

on4 

0157 

oi99 

0242 

0284 

2 

57 

54i7 

546o 

5502 

5545 

5588 

563i 

2 

58 

0327 

o369 

0412 

o454  o497 

o539 

I 

58 

5673 

57i6 

5759 

58o2 

5844 

5887 

I 

59 

o582 

0624 

0667 

O7O9  0752  O794 

O 

59 

593o 

5973 

6oi5 

6o58 

6101 

6i44 

0 

60" 

50" 

40" 

30"  |  20"   10" 

d 

60" 

50" 

40"   30" 

Al7  I  10" 

. 

Ce-tangent  of  41  Degrees. 

Co-tangent  of  40  De^ees. 

s 

P.  Part  J  4   8  13  17  21  25  30  34  38 

v  p  ,<  1"  2"  3"  4"  5"  o'  7"  8"  9" 
ir"{  4   9  •  13  17  2  \  -'u  30  34  38 

LOGARITHMIC    SINES. 


5      £ine  of  50  Degrees. 

d 

Sine  of  51  Degrees. 

•* 

#•    [  10" 

20" 

30" 

40" 

50" 

« 

0" 

10"  1  20" 

30" 

40" 

50" 

0 

9.  8842  54142  7a 

4289 

43o7 

4325 

4342 

59 

o  9.89o5o3 

o52O 

o537 

o554 

o57i 

o588 

59 

T 

436o 

4378 

4395 

44i3 

443i 

4448 

58 

i 

o6o5 

0622 

o639 

o656 

o673 

0690 

58 

2 

4466 

4433 

45oi 

45I9 

4536 

4554 

57 

2 

0707 

0724 

o74i 

o758 

o775 

0-792 

57 

3 

4572 

4589 

46o7 

46254642 

466o 

56 

3 

o8o9 

0826 

o843 

0860 

o877 

0894 

56 

4 

4677 

4695 

47i3 

473o 

4748 

4766 

55 

4 

0911 

0928 

o945 

0962 

°979 

0996 

55 

5    4783 

48oi 

43i8 

4836 

4854 

487i 

54 

5 

ioi3 

io3o 

io47 

1064 

1081 

1098 

54 

6    4889 

49o6 

4924 

494a 

49^9 

4977 

53 

6 

ni5 

Il32 

n49 

1166 

n83 

1200 

53 

7 

4994 

5012 

5o3o 

5o47 

5o65 

5o82 

52 

7 

1217 

1234 

I25l 

1268 

1285 

1302 

52 

8 

5ioo 

5n85i35 

5i53 

Si7o 

5i88 

5i 

8 

1319 

i336 

i353 

i37o 

i387 

i4o4 

5i 

9 

52o5 

5223 

524i 

5258 

5276  5293 

5o 

9 

1421 

i438 

i455 

1472 

i489 

i5o6 

5o 

10 

9-8853H 

5328 

5346 

5364 

5o8i 

5399 

49 

10 

9.891523 

1  54o 

i556 

i573 

i59o 

i6o7 

4o 

ii 

54i6 

5434 

545  1 

5469 

548t> 

55o4 

48 

ii 

1624 

1641 

i658 

i675 

i692 

I7o9 

4-8 

12 

5522 

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58 

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O 

GO" 

50" 

40"  |  30" 

20"  i  10" 

a 

60" 

50"   40" 

30" 

20"   10" 

a 

Co-sine  of  39  Degrees. 

.S 

Co-sine  of  38  Degrees. 

L(  1"  -2"  3' 
r.PartJ  035 

4"  5"  6"  7"  8"  9" 
7   9  10  12  14  16 

.  <  1"  2"  3/!  4"  5"  6"  7"  8"  9" 
P.  Tart  J  2   3   5   7   s  10  12  13  15 

LOGARITHMIC    TANGENT?. 


4 

Tangent  of  50  Degrees. 

c 
a 

Tangent  of  51  Degrees. 

s 

0" 

10" 

20" 

30" 

40" 

50" 

m 

0" 

10" 

20" 

30" 

40" 

50" 

o 

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0 

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O 

60" 

50" 

40" 

30" 

20" 

10" 

c 

60" 

50"   40"   30"   20"   10" 

^ 

Co-tangent  of  39  Degrees. 

.b 

& 

Co-tangent  of  38  Degrees. 

§ 

P  PartJ  l"  2//  3"  4"  5"  G//  7//  8//  °" 
illi  4   9  13  17  21  2G  30  34  39 

P  PartJ  l"  ~'  3"  4"  :V/  °"  7"  8"  9// 
r.iaii^  4   9  13  17  22  26  3Q  35  3p 

J 

7(5 


LOGARITHMIC    SINES. 


a  j      Sine  of  52  Degrees. 

a 
° 

Sine  of  53  Degrees. 

53     o" 

! 

10" 

20'' 

30" 

40" 

50" 

0" 

10"   20" 

30" 

40" 

50" 

09.896532 

6549 

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6598 

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59 

o 

9.9o2349 

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2396 

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2428 

59 

I 

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67i3 

58 

I 

2444 

2460  24?5 

249I 

2507 

2523 

58 

2 

6729 

6746 

6762 

6795 

6812 

57 

2 

2539 

2555 

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2586 

2602 

2618 

57 

3 

6828 

6844 

6861 

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6894 

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56 

3 

2634 

265o 

2666 

2681 

2697 

2713 

56 

4    6926 

6943 

6959 

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55 

4 

2729 

2745 

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2776 

2792 

2808 

55 

5 

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7107 

54 

5 

2824 

2840 

2856 

2871 

2887 

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54 

6 

7123 

7i4o 

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7i89 

7205 

53 

6 

29I9 

2935 

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2966 

2982 

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53 

7 

7222 

7238 

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7287 

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52 

7 

3oi4 

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52 

8 

7320 

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7369 

7385 

7402 

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8 

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9 

74i8 

7434 

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7483 

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9 

32o3 

32I9 

3235 

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3266 

3282 

5o 

IO 

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7533 

7549 

7565 

7582 

7598 

49 

10 

9.9o3298 

33i3 

3329 

3345 

336i 

3377 

49 

ii 

7614 

763i 

7647 

I7663 

7680 

7696 

48 

ii 

3392 

34o8 

3424 

344o 

3455 

48 

12 

7712 

7729 

7745 

7761 

7778 

7794 

47 

12 

3487 

35o3 

35i8 

3534 

355o 

3566 

47 

i3 

7810 

7827 

7843 

7859 

7876 

7892 

46 

1  3 

358i 

3597 

36i3 

3629 

3644 

366o 

46 

i4 
i5 

79o8 
8006 

7924 
8022 

794ij7957 
8o398o55 

7973 
8071 

87ol8° 

45 

44 

i4 
i5 

3676 
377o 

369i 
3786 

37o7 
38o2 

3723 
38i7 

3739 
3833 

3754 
3849 

45 
44 

16 

8104 

8120 

8i368i53 

8i698i85 

43 

16 

3864 

388o 

3896 

39I2 

3927 

3943 

43 

17    8202 

8218 

8234  825o 

82678283 

42 

17 

3959 

3974 

399o 

4oo6 

4021 

4o37 

42 

18 

8299 

83i5 

8332 

!8348 

8364838i 

4i 

18 

4o53 

4o69 

4o84 

4ioo 

4n6 

4i3i 

4i 

19 

8397 

84i3 

84298446 

8462^478 

4o 

:9 

4i47 

4i63 

4i78 

4i94 

4210 

4225 

4o 

20 

Q.  898494 

85n 

8527 

8543 

85598576 

39 

20 

9.9o424i 

4257 

42-72 

4288 

43o4 

43i9 

39 

21 

8592 

8608 

8624 

864i 

86578673 

38 

21 

4335 

435i 

4366 

4382 

4398 

44i3 

38 

22 

8689 

8706 

8722 

8738 

8754877o 

37 

22 

4429 

4445 

446o 

4476 

4492 

45o7 

37 

23 

8787 

88o3 

88i9 

8835 

88528868 

36 

23 

4523 

4539 

4554 

4570 

4586 

46oi 

36 

24 

8884 

89oo 

89i6 

8933 

8949 

8965 

35 

24 

46i7 

4632 

4648 

4664 

4679 

4695 

35 

25 

898i 

8997 

9oi4'9o3o 

9o46 

9o62 

34 

25 

4711 

4726 

4742 

4757 

4773 

4789 

34 

26 

9o78 

9o95 

9m 

9I27 

9i43 

9i59 

33 

26 

48o4 

4820 

4836 

485i 

4867 

4882 

33 

27 

9i76 

9I92 

92O8  9224 

924o 

9256 

32 

27 

4898 

49i4 

4929 

4945 

496o 

4976 

32 

28 

9273 

9289 

93o5l932i 

9337 

9354 

3i 

28 

4992 

5007 

5o23 

5o38 

5o54 

5070 

3i 

29 

937o 

9386 

94O2 

94i8 

9434 

945o 

3o 

29 

5o85 

5ioi 

5n6 

5i32 

5i48 

5i63 

3o 

3o 

9.899467 

9483 

9499;95i5 

953i 

9547 

29 

3o 

9.9o5i79 

5i94 

5210 

5225 

524l 

5257 

29 

3i 

9564 

958o 

9596  96i2 

9628 

9644 

28 

3i 

5272 

5288 

53o3 

53i9 

5334 

535o 

28 

32 

966o 

9677 

969397°9 

9725 

974i 

27 

32 

5366 

538i 

5397 

54  1  2 

5428 

5443 

27 

33 

9757 

9773 

9789 

98o6 

9822 

9838 

26 

33 

5459 

5474 

549o 

55o6 

552i 

5537 

26 

34 

9854 

987o 

9886 

9902 

99i8 

9935 

25 

34 

5552 

5568 

5583 

5599 

56i4 

563o 

25 

35 

995i 

9967 

9983 

9999 

..i5 

24 

35 

5645 

566i 

5676 

5692 

57o8 

5723 

24 

36  9.900047 

oo63 

°°79 

oo96 

0112 

0128 

23 

36 

5739 

5754 

577o 

5785 

58oi 

58i6 

23 

37    oi44 

0160 

oi76|oi92 

0208 

0224 

22 

37 

5832 

5847 

5863 

5878 

5894 

59o9 

22 

38    0240 

0256 

0272 

0289 

o3o5 

0321 

21 

38 

5925 

594o 

5956 

597i' 

5987 

6002 

21 

39    o337 

o353 

o369 

o385 

o4oi 

0417 

2O 

39 

6018 

6o33 

6o49 

6o64 

6080 

6o95 

20 

4o'9.9oo433 

o449 

o465'o48i 

o497 

o5i3 

19 

4o 

9.9o6ni 

6126 

6142 

6157 

6i73 

6188 

I9 

4i 

o529 

o545 

o562 

0578 

o594 

0610 

18 

4i 

6204 

62I9 

6235 

625o 

6265 

6281 

18 

42 

0626 

0642 

o658 

0674 

o69o  0706 

17 

42 

6296 

63i2 

6327 

6343 

6358 

6374 

17 

43 

O722 

0738 

o754 

0770 

0786  0802 

16 

43 

6389 

64o5 

6420 

6436 

645  1 

6466 

16 

44 

O8l8 

o834 

o85o 

0866 

0882 

o898 

i5 

44 

6482 

6497 

65i3 

6528 

6544 

6559 

i5 

45 

o9i4 

o93o 

o946 

0962 

o978 

o994 

i4 

45 

6575 

359o 

66o5 

6621 

6636 

6652 

i4 

46 

IOIO 

1026 

1042 

io58 

1074 

io9o 

i3 

46 

6667 

6683 

6698 

67i3 

6729 

6744 

i3 

47 

1106 

1122 

n38 

n54 

1170 

1186 

12 

47 

6760 

6775 

679i 

6806 

6821 

6837 

12 

48 

I2O2 

1218 

1234 

I25o 

1266 

1282 

II 

48 

6852 

6868 

6883 

6898 

69i4 

6929 

II 

49 

I298 

i3i4 

i33o 

1  346 

i362 

i378 

IO 

49 

6945 

696o 

6975 

699i 

7006 

7O22 

IO 

5o 

9.9oi394 

i4io 

1426 

1  442 

i458 

i474 

9 

5o 

9.9O7o37 

7o52 

7068 

7083 

7°99 

7n4 

9 

5i 

l490 

i5o5 

i5ai 

i537 

r553 

i569 

8 

5i 

7I29 

7i45 

7160 

7175 

7i9i 

7206 

% 

52 

i585 

1601 

1617 

i633  i649 

i665 

7 

52 

7222 

7237 

7252 

7268 

7283 

7298 

7 

53 

1681 

i697 

I7i3 

1729 

1745 

1760 

6 

53 

73i4 

7329 

7344 

736o 

7375 

739i 

6 

54 

i776 

I792 

1808 

1824 

i84o 

i856 

5 

54 

7406 

7421 

7437 

7452 

7467 

7483 

5 

55 

l872 

1888 

i9o4 

I920 

i936 

i95i 

4 

55 

7498 

75i3 

7529 

7544 

7559 

7575 

4 

56 

i967 

i983 

i999 

2Ol5 

2031 

2047 

3 

56 

759° 

76o5 

7621 

7636 

765i 

7667 

3 

57 

2o63 

2079 

2O95 

2110 

2126 

2142 

2 

57 

7682 

7697 

77i3 

7728 

7743 

7759 

2 

58 

2i58 

2174 

2I90 

2206, 

2222 

2238 

I 

58 

7774 

7789 

78o5 

7820 

7835 

785i 

I 

59 

2253 

2269 

2285 

2301 

2317 

2333 

O 

59 

7866 

788i 

7896 

79i-2 

7927 

7942 

O 

60" 

50" 

40'; 

30" 

20" 

10" 

c 

60" 

50' 

40" 

30" 

20" 

10" 

a 

Co-sine  of  3  7 

Degrees. 

§ 

Co-sine  of  36  Degrees. 

i 

f  j//  o"  3"  4' 

5"  6"  7"  8"  9" 

„  T,  A  1"  2"  3"  4"  5"  6"  7"  8"  0"  J 

Part)  2356 

8  10  11  13  15 

I>PartJ  2   3   5   6   8   9  11  12  14  j 

LOGAKiTHMIC      TANGENTS. 


•1 

Tangent  of  52  Degrees. 

A 

Tangent  of  53  Dt  grees. 

35 

0" 

10" 

20" 

30" 

40"  |  50" 

m 

0" 

10" 

20" 

30'   40" 

50" 

0 

10.  107190 

7234 

7277 

7320 

7364 

7407 

59 

0 

10.122886 

2929 

2973 

30*7 

3o6i 

3io5 

59 

I 

745  1 

7494 

7537 

758i 

7624 

7668 

58 

i 

3i48 

3192 

3236 

3280 

3324 

3368 

58 

2 

7711 

7754 

7798 

784i 

7885 

7928 

57 

2 

34n 

3455 

3499 

3543 

3587 

363o 

57 

3 

7972 

8oi5 

8o58 

8102 

8i45 

8!89 

56 

3 

3674 

37i8 

3762 

38o6 

385o 

3893 

56 

4 

8-232 

8275 

83i9 

8362 

84o6 

8449 

55 

4 

3937 

398i 

4o25 

4069 

4n3 

4i57 

55 

5 

8493 

8536 

8579 

8623 

8666 

8710 

54 

5 

4200 

4244 

4288 

4332 

4376 

4420 

54 

6 

6753 

8797 

884o 

8884 

8927 

8970 

53 

6 

4463 

4507 

455i 

4595 

4639 

4683 

53 

7 

9014 

9o57 

9101 

9i44 

9188 

9231 

52 

7 

4727 

477° 

48i4 

4858 

4902 

4946 

52 

8 

9275 

93i8 

936i 

94o5 

9448 

9492 

5i 

8 

499o 

5o34 

5077 

5l2I 

5i65 

5209 

5i 

9 

9535 

9579 

9622 

9666 

9709 

9753 

5o 

9 

5253 

5297 

534i 

5385 

5428 

5472 

5o 

1C 

10.  109796 

984o 

9883 

9926 

9970 

,.i3 

49 

10 

10.  I255i6 

556o 

56o4 

5648 

5692 

5736 

49 

ii 

io.noo57 

OIOO 

oi44 

0187 

023l 

0274 

48 

ii 

578o 

5823 

5867 

0911 

5955 

5999 

48 

12 

o3i8 

o36i 

o4o5 

o448 

0492 

o535 

47 

12 

6o43 

6087 

6i3i 

6175 

6219 

6262 

47 

i3 

0579 

0622 

0666 

0709 

0752 

0796 

46 

i3 

63o6 

635o 

6394 

6438 

6482 

6526 

46 

i4 

0839 

o833 

0926 

0970 

ioi3 

io57 

45 

i4 

6570 

66i4 

6658 

6701 

6745 

6789 

45 

i5 

IIOO 

n44 

1187 

I23l 

1274 

i3i8 

44 

i5 

6833 

6877 

6921 

6965 

7009 

7o53 

44 

16 

i36i 

i4o5 

1  448 

1492 

i535 

l579 

43 

16 

7°97 

7141 

7i85 

7229 

7273 

73i6 

43 

J7 

1622 

1666 

1709 

i753 

1797 

1840 

42 

17 

736o 

74o4 

7448 

7492 

7536 

758o 

42 

18 

i884 

1927 

1971 

2Ol4 

2o58 

2101 

4i 

18 

7624 

7668 

7712 

7756 

7800 

7844 

4i 

*9 

2i45 

2188 

2232 

2275 

2319 

2362 

4o 

19 

7888 

7932 

7976 

8020 

8o63 

8107 

4o 

20 

10.  112406 

2449 

2493 

2536 

258o 

2623 

39 

20 

10.  I28i5i 

8195 

8239 

8283 

8327 

8371 

39 

21 

2667 

2711 

2754 

2798 

2841 

2885 

38 

21 

84i5 

8459 

85o3 

8547 

859i 

8635 

38 

22 

2928 

2972 

3oi5 

3o59 

3lO2 

3i46 

37 

22 

8679 

8723 

8767 

8811 

8855 

8899 

37 

23 

3i89 

3233 

3277 

3320 

336/v 

34o7 

36 

23 

8943 

8987 

9o3i 

9o75 

9119 

9i63 

36 

24 

345i 

3494 

3538 

358i 

3625 

3669 

35 

24 

9207 

925i 

9295 

9339 

9383 

9427 

35 

25 

3712 

3756 

3799 

3843 

3886 

393o 

34 

25 

947i 

95i5 

9559 

96o3 

9647 

9691 

34 

26 

3974 

4017 

4o6i 

4io4 

4i48 

4191 

33 

26 

9735 

9779 

9823 

9867 

9911 

9955 

33 

27 

4235 

4279 

4322 

4366 

4409 

4453 

32 

27 

9999 

..43 

..87 

.i3i 

.I75 

.219 

32 

28 

4496 

454o 

4584 

4627 

467i 

47i4 

3i 

28 

io.i3o263 

0307 

o35i 

oSgS 

o439 

o483 

3  1 

29 

4758 

4802 

4845 

4889 

4932 

4976 

3o 

29 

o527 

o57i 

o6i5 

o659 

0703 

0747 

3o 

3o 

10.  116020 

5o63 

5107 

5i5o 

5i94 

5238 

29 

3o 

10.  i3o79i 

o835 

0879 

0923 

o967 

IOII 

29 

?l 

5281 

5325 

5368 

54i2 

5456 

5499 

28 

3i 

io55 

1099 

n43 

1187 

1232 

1276 

28 

32 

5543 

5586 

563o 

5674 

5717 

576i 

27 

32 

l320 

1  364 

i4o8 

i452 

i496 

1  54o 

27 

33 

58o4 

5848 

5892 

5935 

5979 

6o23 

26 

33 

1  584 

1628 

1672 

1716 

1760 

1804 

26 

34 

6066 

6110 

6i53 

6197 

6241 

6284 

25 

34 

1  848 

1892 

1936 

1981 

2025 

2069 

25 

35 

6328 

6372 

64i5 

64Si) 

65o2 

6546 

24 

35 

2Il3 

2157 

2201 

2245 

2289 

2333 

24 

36 

6590 

6633 

6677 

6721 

6764 

6808 

23 

36 

2377 

2421 

2465 

2509 

2554 

2598 

23 

3? 

6852 

6895 

6939 

6962 

7026 

7070 

22 

37 

2642 

2686 

2730 

2774 

2818 

2862 

22 

38 

7n3 

7i57 

7201 

7244 

7288 

7332 

21 

38 

29o6 

2950 

2994 

3o39 

3o83 

3127 

21 

39 

7375 

7419 

7463 

75o6 

755o 

7594 

20 

39 

3i7i 

32i5 

3259 

33o3 

3347 

339i 

20 

4o 

10.  117637 

7681 

7725 

7768 

7812 

7856 

J9 

4o 

10.  133436 

348o 

3524 

3568 

36i2 

3656 

19 

4i 

7899 

7943 

7987 

8o3o 

8o74 

8118 

18 

4i 

37oo 

3744 

3789 

3833 

3877 

3921 

18 

42 

8161 

82o5 

8249 

8292 

8336 

838o 

ll 

42 

3965 

4009 

4o53 

4097 

4i42 

4i86 

17 

43 

8423 

8467 

85u 

8555 

8598 

8642 

16 

43 

4230 

4274 

43i8 

4362 

44o6 

445  1 

16 

44 

8686 

8729 

8773 

8817 

8860 

8904 

i5 

44 

4495 

4539 

4583 

4627 

467i 

47i5 

i5 

45 

8948 

8992 

9o35 

9079 

9123 

9166 

i4 

45 

476o 

48o4 

4848 

4892 

4936 

4980 

i4 

46 

9210 

9254 

9297 

934i 

9385 

9429 

i3 

46 

5o25 

5069 

5ii3 

5i57 

5201 

5245 

i3 

4? 

9472 

95i6 

956o 

96o3 

9647 

9691 

12 

47 

529o 

5334 

5378 

5422 

5466 

55io 

12 

48 

9735 

9778 

9822 

0866 

9909 

9953 

II 

48 

5555 

5599 

5643 

5687 

573i 

5775 

II 

49 

9997 

..4i 

..84 

.128 

.172 

.216 

10 

49 

5820 

5864 

5908 

5952 

5996 

6o4i 

10 

5o 

10.120259 

o3o3 

o34? 

o39i 

o434 

0478 

9 

5o 

io.i36o85 

6129 

6173 

6217 

6262 

63o6 

? 

5i 

O522 

o565 

0609 

o653 

0697 

0740 

8 

5i 

635o 

6394 

6438 

6483 

6527 

657i 

8 

52 

0784 

0828 

0872 

0915 

°959 

ioo3 

7 

52 

66i5 

6659 

6704 

6748 

6792 

6836 

7 

53 

1047 

1091 

n34 

1178 

1222 

1266 

6 

53 

6881 

6925 

6969 

7013 

7057  7102 

6 

54 

1  309 

i353 

i397 

i44i 

i484 

i528 

5 

54 

7i46 

7190 

7234 

7279 

7323  7367!  * 

55 

1572 

1616 

i659 

I7o3 

1747 

1791 

4 

55 

74n 

7455 

75oo 

7544 

758817632 

4 

56 

i835 

1878 

1922 

1966 

2010 

2o53 

3 

56 

7677 

7*721 

7765 

7809 

78547898 

3 

57 

2097 

2l4l 

2i85 

2229 

2272 

23i6 

2 

57 

7942 

7986 

8o3i 

8075 

8n98i63 

2 

58 

236o 

2404 

2448 

2491 

2535 

2579 

I 

58 

8208 

8252 

8296 

834i 

8385!8429 

I 

59 

2623 

2667 

2710 

2754 

2798 

2842 

0 

59 

8473 

85i8 

8562 

8606 

865o8695 

O 

60"      50"   40" 

30" 

20"  |  10" 

. 

60" 

50" 

40" 

30' 

20"   10" 

"." 

Co-tangent  of  37  Degrees. 

& 

Co-tangent  of  36  Degrees. 

1 

P  Por-tJ  *"   2"   3//   4//   5//  6"   7"   8"   9" 

<  1"  2"  3"  4"  5"  6"  7"  8"  9" 

irl{  1   9  13  17  22  26  31  35  39 

Lrt{  4   9  13  18  22  26  31  35  40 

78 


LOGARITHMIC    SINES. 


c 

Sine  of  54  Degrees. 

a 

Sine  of  55  Degrees. 

s 

0" 

10" 

20"  [  30"   40" 

50" 

Ii 

0"      10" 

20" 

30" 

40" 

50" 

0 

9.9o7958 

7973 

•7988  80048019 

8o34 

69 

o 

9-913365 

33.70 

3394 

3409 

3423 

3438 

59 

I 

8049 

8o65 

808080958111 

8126 

58 

I 

3453 

3468 

34S2 

3497 

35i2 

3527 

55 

2 

8i4i 

8i56 

8x7381878202 

8217  57 

2 

354i 

3556 

357i 

3585 

36oo 

36i5 

5- 

3 

8233 

8248 

826382798294 

8309 

56 

3 

363o 

3644 

3659 

3674 

3688 

37o3 

50 

4 

8324 

834o83558370:8385 

84oi 

55 

4 

37i8 

3733 

3747 

3762 

3777 

379i 

55 

5 
6 

84i6 
85o7 

843i  844684628477 
8523  8538  8553!8568 

8492 
8584 

54 
53 

5 
6 

38o6 

3894 

382i 
39o9 

3836 

385o 
3938 

3865 
3953 

388o 
3968 

54 
53 

8 
9 

8599 
8690 
878i 

8614862986448660 
87068721  8736875i 
8797881388378843 

8675 
8766 
8857 

52 

5i 
5o 

8 
9 

3982 
4o7o 
4i58 

3997 
4o85 
4i73 

4012 
4ioo 

4i88 

4026 

4n4 
4202 

4o4i 
4129 

4217 

4o56 

4i44 

4232 

52 

5i 

5o 

JO 

9.9o8872 

8888890389188933 

8949 

49 

IO 

9.914246 

4261 

42-76 

4290 

43o5 

4320 

49 

ii 

8964 

8979  8994  9009  9025 

9040 

48 

ii 

4334 

4349 

4364l4378 

4393 

44o7 

48 

12 

9o55 

9o7o!9o85 

9101  9116 

9i3i 

47 

12 

4422 

44?7 

445  1 

4466 

448  1 

4495 

47 

l2 

9i46 

9161  9i76 

9192  9207 

9222 

46 

i3 

45io 

4524 

4539 

4554 

4568 

4583 

46 

i4 

9237 

9252  9267 

9283  9298 

93!3 

45 

i4 

4598 

4612 

4627 

464i 

4656 

467i 

45 

i5 

9328 

93439358 

93749389 

94o4 

44 

i5 

4685 

4700 

4714 

4729 

4744 

4758 

44 

16 

9419 

94349449 

9464  948o 

9495 

43 

16 

4773 

4787 

4802 

48i7 

483i 

4846 

43 

i7 

9525  9540 

9555 

957° 

9586 

42 

i7 

486o 

4875 

489o 

4904 

4919 

4933 

42 

18 

9601 

96169681 

96469661 

9676 

4i 

18 

4948 

4962 

4977 

4992 

5oo6 

5021 

4i 

19 

9691 

97079722 

9737i9752 

9767 

4o 

19 

5o35 

5o5o 

5o64 

5o79 

5094 

5io8 

4o  ' 

20 

9.9o9782 

97979812 

9827 

9843 

9858 

39 

20  9.9l5l23 

5i37 

5i52 

5i66 

5i8i 

5i96 

39 

21 

9873 

9888^903 

9918 

9933 

9948 

38 

21 

52IO 

5225 

5239 

5254 

5268 

5283 

38 

22 

9963 

9978  9994 

...9 

..24 

..39 

37 

22 

5297 

53i2 

5326 

534i 

5356 

537o 

37 

23 

9.910054 

0069  008^ 

0099 

on4 

0129 

36 

23 

5385 

5399 

54i4 

5428 

5443 

5457 

36 

24 

oi44 

oi5q  0175 

0190 

0205 

0220 

35 

24 

5472 

5486 

55oi 

55i5 

553o 

5544 

35 

25 

0235 

O25o  O265 

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0295 

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25 

5559 

5573 

5588 

56o2 

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563i 

34 

26 

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26 

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27 

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0476 

0491 

32 

27 

5733 

5747 

5762 

5776 

579i 

58o5 

32 

28 

o5o6 

o52i  o536 

o55i 

o566 

o58i 

3i 

28 

5820 

5834 

5849 

5863 

5878 

5892 

3i 

29 

o596 

0611  0626 

o64i 

o656 

0671 

3o 

29 

59o7 

5921 

5g36 

595o 

5965 

5979 

3o 

3o 

9.910686 

0701  0716 

0731 

o746 

0761 

29 

3o 

9.915994 

6008 

6o23 

6o37 

6o52 

6066 

29 

3i 

o776 

0791  0806 

0821 

o836 

o85i 

28 

3i 

6081 

6095 

6109 

6124 

6i38 

6i53 

28 

32 

0866 

0881  0896 

0911 

0926 

0941 

27 

32 

6i67 

6182 

6196 

6211 

6225 

6240 

27 

33 

0956 

0971 

0986 

IOOI 

1016 

io3i 

26 

33 

6254 

6268 

6283 

629-7 

63i2 

6326 

26 

34 

io46 

1061 

1076 

1091 

1106 

II2I 

25 

34 

634i 

6355 

6369 

6384 

6398 

64i3 

25 

35 

n36 

n5i 

1166 

1181 

1196 

I2II 

24 

35 

6427 

6442 

6456 

647o 

6485 

6499 

24 

36 

1226 

1241 

1256 

1271 

1286 

i3oo 

23 

36 

65i4 

6528 

6543 

6557 

657i 

6586 

23 

37 

i3i5 

i33o 

1  345 

i36o 

i375 

1390 

22 

37 

6600 

66i5 

6629 

6643 

6658 

6672 

22 

38 

i4o5 

1420 

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i45o 

i465 

i48o 

21 

38 

6687 

67o: 

67i5 

673o 

6744 

o759 

21 

39 

1496 

i5io 

i525 

i54o 

i555 

1569 

20 

39 

6773 

6787 

6802 

6816 

683o 

6845 

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4o 

9.9n584 

1  599 

i6i4 

1629 

1  644 

1659 

1  9 

4o 

9.916859 

6874 

6888 

6902 

6917 

693i 

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4i 

1674 

1689 

1704 

1719 

1734 

i748 

18 

4i 

6946 

6960 

6974 

6989 

7°o3 

7oi7 

1  8 

42 

1763 

i778 

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1808 

1823 

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7o46 

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17 

43 

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1897 

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16 

43 

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7i75 

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16 

44 

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1972 

1987 

2OO2 

20I7 

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44 

7204 

72l8 

7233 

7247 

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7276 

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45 

2031 

2046 

2061 

2076 

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45 

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73o4 

73i9 

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7362 

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46 

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2i36 

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2i65 

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46 

7376 

739o 

74o5 

7419 

7433 

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47 

22IO 

2255 

2269^284 

12 

47 

7462 

7476 

749i 

75o5 

75l9 

7534 

12 

48 

2299 

23~i4  2329 

2344 

33583373 

II 

48 

7548 

7562 

7576 

759i 

76o5 

7619 

II 

49 

2388 

24o3  24  1& 

2433 

2448  2462 

IO 

49 

7634 

7648 

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7677 

7691 

77o5 

10 

5o 

9.912477 

2492  25o7 

2522 

25372551 

9 

5o 

9-9I77I9 

7734 

7748  7762 

7777 

7791 

9 

5i 

2566 

258i 

2596 

26ll 

2625  2640 

8 

5i 

78o5 

7819 

7834 

7848 

7862 

7877 

8 

52 

2655 

2670 

2685 

2700 

2714  2729 

7 

52 

•7891 

79°5 

-7919 

7934 

7948 

7962 

7 

53 

2744 

275g 

2774 

2788 

28033818 

6 

53 

7976 

•7991  8oo5  8019 

8o33 

8o48 

6 

54 

2833 

2848 

2862 

2877 

2892  2907 

5 

54 

8062 

8o76  8090  8io5 

81198133 

5 

,  55 

2922 

2936 

2g5i 

2966 

2981  2995 

4 

55 

8i47 

8i628i76'8i9o 

8204  8219 

4 

56 

3oio 

3o25 

3o4o 

3o55 

3069  3o84 

3 

56 

8233182478261  8276  82908304  Z 

57 

3099 

3n4 

3i283i43 

3i583i73 

2 

57 

83i8  8333  8347  836i83758389  2 

58 

3187 

•  202 

3217  3232 

324?  326i 

I 

58 

84o4  84i8  8432  8446  846i  8475  i 

59 

3276 

3291 

33o6  3320  3335335o  o 

59 

848985o385i7  853s  8546856o  o 

60" 

50" 

40"  I  30"   20"   10"  |  .  1 

60"    |  50"   40"   30"   20"   10-'' 

Co-sine  of  35  Degrees. 

Co-sine  of  34  Degrees. 

P  P,rt5  l'  -'  3"  4"  5"  6"  7"  8"  9"       .11"  2"  3"  4"  5"  6"  7"  8"  9" 

iri)  2   3   5   6   8   9  11  12  14  ij  F'FartS  1   3   4   6   7   9  10  12  13 

LOGARITHMIC    TANGENTS. 


d 

Tangent  of  54  Degrees. 

c     Tangent  of  55  Degrees.    j 

9 

0" 

10"  |  20" 

30" 

40" 

50" 

JK 

0"     10" 

20" 

30" 

40" 

50" 

o 

io.i38739 

8783 

8828 

8872 

89i6 

896o 

59 

0 

10.  i54773 

48i8 

4863 

49o8 

4952 

4997 

V 

I 

9005 

9049 

9o93 

9i38 

9l82 

9226 

58 

i 

5o42 

5o87 

5i32 

5i77 

5221 

5266 

58 

1   2 

9270 

93i5 

9359 

94o3 

9448 

9492 

57 

2 

53n 

5356 

54oi 

5446 

549o 

5535 

57 

3 

9536 

958o 

962$ 

9669 

97i3 

9758 

56 

3 

558o 

5625 

567o 

57i5 

5759 

58o4 

56 

x 

9802 

9846 

989i 

9935 

9979 

:.24 

55 

4 

5849 

5894 

5939 

5984 

6o29 

6o73 

55 

5 

10.  i4oo68 

0112 

oi57 

O2OI 

r  r 

0245 

O29O 

54 

5 

6118 

6i63 

6208 

6253 

6298 

6343 

54 

6 

o334 

o378 

042  3 

0467 

o5n 

o556 

53 

6 

6388 

6432 

6477 

6522 

6567 

6612 

53 

7 

0600 

o644 

o689 

0733 

0777 

0822 

52 

7 

6657 

6702 

6747 

679i 

6836 

6881 

52 

8 

0866 

0910 

o955 

0999 

io43 

1088 

5i 

8 

6926 

69-71 

7016 

7o6i 

7io6 

7161 

5i 

9 

Il32 

1176 

1221 

1265 

i3o9 

i354 

5o 

9 

7i95 

7240 

7285 

733o 

7375 

7420 

5o 

10 

10.141398 

1442 

148? 

i53i 

i576 

1620 

49 

10 

10.  i57465 

75io 

7555 

76oo 

7645 

7689 

49 

1  1 

i664 

1709 

i753 

1797 

1842 

1886 

48 

ii 

7734 

7779 

7824 

7869 

79i4 

7959 

48 

12 

ig3i 

1975 

2OI9 

2064 

2108 

2l52 

47 

12 

8oo4 

8049 

8o94 

8i39 

8i84 

8229 

47 

i3 

2197 

2241 

2286 

233o 

2374 

24l9 

46 

i3 

8273 

83i8 

8363 

84o8 

8453 

8498 

46 

i4 

2463 

25o8 

2552 

2596 

2641 

2685 

45 

i4 

8543 

8588 

8633 

8678 

8723 

8768 

45 

i5 

2730 

2774 

2818 

2863 

2907 

2952 

44 

i5 

88i3 

8858 

89o3 

8948 

8993 

9o38 

44 

16 

2996 

3o4i 

3o85 

3l29 

3174 

3218 

43 

16 

9o83 

9128 

9i73 

92l8 

9263 

93°7 

43 

i? 

3263 

3307 

335i 

3396 

344o 

3485 

42 

i? 

9352 

9397 

9442 

9487 

9532 

9577 

42 

18 

3529 

3574 

36i8 

3662 

37o7 

375i 

4i 

18 

9622 

9667 

9712 

9757 

98o2 

9847 

4i 

J9 

3796 

384o 

3885 

3929 

3974 

4oi8 

4o 

T9 

9892 

9937 

9982 

..27 

..72 

.117 

4o 

20 

10.  i44o<32 

4107 

4i5i 

4196 

4240 

4285 

39 

20 

10.  160162 

O2O7 

O252 

0297 

o342 

o387 

39 

21 

4329 

4374 

44i8 

4463 

45o7 

455i 

38 

21 

0432 

o477 

O522 

o567 

0612 

o657 

38 

22 

4596 

464o 

4685 

4729 

4774 

4818 

37 

22 

0703 

o748 

o793 

o838 

o883 

O928 

37 

23 

4863 

4907 

4952 

4996 

5o4i 

5o85 

36 

23 

0973 

1018 

io63 

1108 

ii53 

1198 

36 

24 

5i3o 

5174 

5219 

5263 

53o8 

5352 

35 

24 

1243 

1288 

i333 

i378 

1423 

i468 

35 

25 

5397 

544  1 

5486 

553o 

5575 

56i9 

34 

25 

i5i3 

i558 

i6o3 

1  648 

i693 

i739 

34 

26 

5664 

5708 

5753 

5797 

5842 

5886 

33 

26 

1784 

1829 

i874 

1919 

i964 

2009 

33 

27 

593i 

5975 

6020 

6064 

6io9 

6i53 

32 

27 

20  54 

2099 

2i44 

2189 

2234 

2279 

32 

28 

6198 

6242 

6287 

633i 

6376 

6420 

3i 

28 

2325 

2370 

24i5 

2460 

25o5 

2550 

3i 

29 

6465 

6509 

6554 

6598 

6643 

6687 

3o 

29 

2595 

2640 

2685 

273o 

2775 

2821 

3o 

3o 

10.146732 

6777 

6821 

6866 

69io 

6955 

29 

3o 

10.162866 

2911 

2956 

3ooi 

3o46 

3o9i 

29 

3i 

6999 

7o44 

7088 

7i33 

7i77 

7222 

28 

3i 

3i36 

3i8i 

3227 

3272 

33i7 

3362 

28 

32 

7267 

73n 

7356 

74oo 

7445 

7489 

27 

32 

34o7 

3452 

3497 

3542 

3588 

3633 

27 

33 

7534 

7578 

7623 

7668 

77I2 

7757 

26 

33 

3678 

3723 

3768 

38i3 

3858 

39o4 

26 

34 

7801 

7846 

7890 

7935 

798o 

8024 

25 

34 

3949 

3994 

4o39 

4o84 

4i29 

4i74 

25 

35 

8069 

8n3 

8i58 

8203 

8247 

8292 

24 

35 

4220 

4265 

43io 

4355 

44oo 

4445 

24 

36 

8336 

838i 

8425 

8470 

85i5 

8559 

23 

36 

449i 

4536 

458! 

4626 

467i 

47i6 

23 

•37 

8604 

8648 

8693 

8738 

8782 

8827 

22 

37 

4762 

48o7 

4852 

4897 

4942 

4988 

22 

38 

8871 

8916 

8961 

9oo5 

9o5o 

9o95 

21 

38 

5o33 

5o78 

5i23 

5i68 

52i3 

5259 

21 

39 

9i39 

9184 

9228 

9273 

93i8 

9362 

20 

39 

53o4 

5349 

5394 

5439 

5485 

553o 

2O 

4o 

10.  149407 

9452 

9496 

954i 

9585 

963o 

'9 

4o 

io.i65575 

5620 

5666 

57n 

5756 

58oi 

19 

4i 

9675 

9719 

9764 

9809 

9853 

9898 

18 

4i 

5846 

5892 

5937 

5982 

6o27 

6o73 

1  8 

42 

9943 

9987 

..32 

..76 

.121 

.166 

J7 

42 

6118 

6i63 

6208 

6253 

6299 

6344 

*7 

43 

IO.l5o2IO 

0255 

o3oo 

o344 

o389 

o434 

16 

43 

6389 

6434 

648o 

6525 

657o 

66i5 

16 

44 

0478 

o523 

o568 

0612 

o657 

O7O2 

i5 

44 

6661 

67o6 

675i 

6796 

6842 

6887 

i5 

45 

0746 

0791 

o836 

0880 

0926 

o97o 

i4 

45 

6932 

6977 

7023 

7o68 

7n3 

7i58 

i4 

46 

1014 

1059 

no4 

1149 

n93 

12^8 

i3 

46 

7204 

7249 

7294 

734o 

7385 

743o 

i3 

47 

1283 

1327 

1372 

1417 

i46i 

i5o6 

12 

47 

7475 

752I 

7566 

76ii 

7657 

7702 

12 

48 

i55i 

iSgS 

i64o 

i685 

i73o 

i774 

II 

48 

7747 

7792 

7838 

7883 

7928 

7974 

II 

49 

1819 

1  864 

1908 

1953 

i998 

2043 

10 

49 

8019 

8o64 

8io9 

8i55 

8200 

8245 

10 

5o 

10.152087 

2l32 

2177 

2221 

2266 

23ll 

9 

5o 

10.  168291 

8336 

838i 

8427 

8472 

85i7 

9 

5i 

2356 

2400 

2445 

2490 

2535 

2579 

8 

5i 

8563 

8608 

8653 

8699 

8744 

8789 

8 

52 

2624 

2669 

2713 

2758 

28o3 

2848 

7 

52 

8835 

8880 

8925 

897i 

9oi6 

9o6i 

7 

53 

2892 

2937 

2982 

3027 

3o7i 

3n6 

6 

53 

9io7 

9l52 

9i97 

9243 

9288 

9333 

6 

54 

3i6i 

3206 

3a5o 

3295 

334o 

3385 

5 

54 

9379 

9424 

9469 

95i5 

956o 

96o5 

5 

55 

343o 

3474 

35i9 

3564 

36o9 

3653 

4 

55 

965i 

9696 

9742 

9787 

9832 

9878 

4 

56 

3698 

3743 

3788 

3832 

3877 

3922 

3 

56 

9923 

9968 

..i4 

..59 

.io5 

.i5o 

3 

57 

3967 

4012 

4o56 

4ioi 

4i46 

4i9i 

2 

57 

10.  I7oi95 

0241 

0286 

o33i 

o377 

0422 

2 

58 

4236 

4280 

4325 

437o 

44i5 

446o 

I 

58 

o468 

o5i3 

o558 

o6o4 

o649 

o695 

I 

59 

45o4 

4549 

4594 

4639 

4684 

4728 

0 

59 

o74o 

o785 

o83i 

o876 

O922 

o967 

O 

•50" 

50"  |  40" 

30" 

20" 

10" 

00"    |  50" 

40" 

30"  |  20" 

10" 

. 

Co-tangent  of  35  Degrees. 

a 

Co-tangent  of  34  Degrees. 

•S 
S 

P  P-,rf  S  l"   2"   3"   4"   5//   6"   7//   8//   9// 

$  4   9  13  18  22  27  31  36  40 

p  p  .(!"  2"  3"  4"  5"  6"  7"  8"  9" 
\  5   9  14  18  23  27  32  36  11 

LOGARITHMIC    SINES. 


1 

d 

Sine  of  56  Degrees. 

_g 

Sine  of  57  Degrees. 

m 

G' 

10" 

20" 

30'' 

40"   50" 

§ 

0" 

10" 

20" 

30" 

40" 

50" 

0 

;.9i8574 

8588 

86o3 

86i7 

863i 

8645 

59 

O 

9.92359i 

36o5 

36i9 

3632 

3646 

366o 

59 

i 

8659 

8674 

8688 

8702 

87i6 

873o 

58 

I 

3673 

3687 

3701 

37i4 

372S 

3742 

58 

2 

8745 

8759 

8773 

8787 

8801 

88i5 

57 

2 

3755 

3769 

3783 

3796 

38io 

3824 

57 

3 

883o 

8844 

8858 

8872 

8886 

89oo 

56 

3 

3837 

385i 

3865 

3878 

3892 

39o6l56 

4 

89i5 

8929 

8943 

8957 

897i 

8985 

55 

4 

39i9 

3933 

3946 

396o 

3974 

3987-, 

c 

oooo 

9oi4 

9028 

9o42 

9o56 

9°7° 

54 

5 

4ooi 

4oi5 

4028 

4042 

4o55 

4069 

54 

6 

9o85 

9°99 

9n3 

9I27 

9i4i 

9i55 

53 

6 

4o83 

4o96 

4no 

4124 

4i37 

4i5i 

53 

7 

9l69 

9i84 

9i98 

92I2 

9226 

9240 

52 

7 

4i64 

4178 

4192 

42o5 

42I9 

4232 

02 

8 

9254 

9268 

9282 

9297 

93n 

9325 

5i 

8 

4246 

4260 

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0 

60"     50" 

40" 

30"   20" 

10" 

Q 

60" 

50" 

40" 

30" 

20" 

10" 

d 

Co-sine  of  33  Degrees. 

Co-sine  of  32  Degrees. 

1 

^\"  2"  3"  4"  5"  6"  7"  8"  9" 
l.lartj  x   3   4   6   7   8  10  11  13 

v  v  A  I1  2"  3"  4"  5"  6"  7"  8"  9" 
P.PartJ  !   3   4   5   7   8   9  11  12 

LOGARITHMIC    TANG  E  NT  d. 


81 


d 

Tangent  of  56  Degrees. 

i 

Tangent  of  57  Degrees. 

34 

0" 

10" 

20" 

30" 

40"  ;  i»0" 

* 

0"     |  10"  |  20-'' 

30" 

40" 

50" 

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54 

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2 

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3930 

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O 

60" 

50"  |  40" 

30" 

20"   10" 

60" 

50" 

40" 

30" 

20" 

10" 

. 

Co-tangent  of  33  Degrees. 

Co-tangent  of  32  Degrees. 

1 

1»  P*rt$  l"  2"  3//  4"  5"  6"  7"  8"  9// 

C  1"  2"  3"  4"  5"  6"  7"  8"  9" 

{  5   9  14  18  23  27  32  37  41 

ir  )  5   9  14  19  23  28  33  37  42 

82 


LoGARITIIMiO     SlNES. 


1  — 
.3 

Sine  of  58  Degrees. 

d 

a 

Sine  of  59  Degrees. 

'  — 

55 

0" 

10" 

20"   30"   40"   50" 

SI 

0"    [  10" 

20"  |  30" 

40" 

50" 

0 

9.928420 

8434 

8447  846o  8473 

8486;59 

o  9.933066 

3078 

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3io4 

3n6 

3l29 

59 

I 

8499 

85i3 

8526  8539  8552 

856558 

i 

3i4i 

3i54 

3i67 

3179 

3i92 

32o5 

58 

2 

8678 

859i 

86o586i8863i 

8644 

57 

2 

32I7 

323o 

3243 

3255 

3268 

328o 

57 

3 

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8723 

56 

3 

3293 

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33i8 

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3344 

3356 

/ 

56 

4 

8736 

8749 

8762  8775  8788 

8801 

55 

4 

3369 

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3394 

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55 

c 

88i5 

8828 

884i 

8854  8867 

8880 

54 

5 

3445 

3457 

347o 

3482 

3495 

35o8 

54 

6 

8893 

8906  89i9 

8933  8946 

8959 

53 

6 

3520 

3533 

3545 

3558 

357i 

3583 

53 

7 

8972 

8985 

8998 

9on 

9024 

9o37 

52 

7 

3596 

36o8 

362i 

3633 

3646 

3659 

52 

8 

9o5o 

9o63  9o77 

9o9o  9io3 

9-116 

5i 

8 

367i 

3684 

3696 

37°9 

3722 

3734 

5i 

9 

9129 

9142 

9i55 

91689181 

9194 

5o 

9 

3747 

3759 

3772 

3784 

3797 

38io 

5o 

10 

9-929207 

92209233  9247 

9260 

9273 

49 

10 

9.933822 

3835 

3847 

386o 

3872 

3885 

49 

ii 

9286 

9299 

93129325 

9338 

g35i 

48 

ii 

3898 

39io 

3923 

3935 

3948 

396o 

48 

12 

9364 

9377939o|94o3 

9416 

9429 

47 

12 

3973 

3985 

3998 

4on 

4o23 

4o36 

47 

i3 

9442 

9455  9468  9482 

9495 

ODOO 

46 

i3 

4o48 

4o6i 

4o73 

4o86 

4o98 

4m 

46 

i4 

9521 

9547  956o 

9573 

9586 

45 

i4 

4l23 

4i36 

4i48 

4i6i 

4i74 

4i86 

45 

i5 

9599 

9612  9625  9638 

965i 

9664 

44 

i5 

4199 

4211 

4224 

4236 

4249 

4261 

44 

16 

9677 

9690  97o3 

9716 

9729 

9742 

43 

16 

4274 

4286 

4209 

43n 

4324 

4336 

43 

I7 

9755 

9768978i 

9794 

98o7 

9820 

42 

i7 

4349 

436i 

4374 

4386 

4399 

44n 

42 

18 

9833 

9846|9859 

9872 

9885 

9898 

4i 

18 

4424 

4436 

4449 

446i 

4474 

4486 

4i 

i9 

9911 

99249937 

995o 

9963 

9976 

4o 

19 

4499 

45n 

4524 

4536 

4549 

456i 

4o 

20 

9.929989 

.  .  .2 

.  .i5 

..28 

..54 

39 

20 

9<934574 

4586 

4599 

46n 

4624 

4636 

39 

21 

9.93oo67 

0080  oo93 

0106 

on9 

Ol32 

38 

21 

4649 

466i 

4674 

4686 

4699 

47n 

38 

22 

oi45 

oi58  oi7i 

0184 

oi97 

O2  1  0 

37 

22 

4723 

4736 

4748 

4761 

4773 

4786 

37 

23 

0223 

0236  0249 

0262 

0274 

028-7 

36 

23 

4798 

48n 

4823 

4836 

4848 

486i 

36 

24 

o3oo 

o3i3o326 

0339 

o352 

o365 

35 

24 

4873 

4885 

4898 

49io 

4923 

4935 

35 

25 

o378 

o39i 

o4o4 

0417 

o43o 

o443 

34 

25 

4948 

496o 

4973 

4985 

4997 

5oio 

34 

26 

o456 

04690482 

o495 

o5o7 

o52O 

33 

26 

5022 

5o35 

5o6o 

5o72 

5o84 

33 

27 

o533 

o546o559 

0572 

o585 

o598 

32 

27 

5o97 

5io9 

5l22 

5i34 

5i47 

5i59 

32 

28 

0611 

0624  o637 

o65o 

o663 

o675 

3i 

28 

5i7i 

5i84 

5i96 

52O9 

5221 

5234 

3i 

29 

0688 

O7OI 

o7i4 

0727 

o74o 

o753 

3o 

29 

5246 

5258 

527I 

5283 

5296 

53o8 

3o 

3o 

9.93o766 

°779  °792 

o8o4 

o8i7 

o83o 

29 

3o 

9.935320 

5333 

5345 

5358 

537o 

5382 

29 

3i 

o843 

o856o869 

0882 

o895 

0908 

28 

3i 

5395 

5407 

5420 

5432 

5444 

5457 

28 

32 

0921 

o933 

o946 

o959 

o972 

o985 

27 

32 

5469 

5482 

5494 

55o6 

55i9 

553i 

27 

33 

0998 

IOII 

1024 

io36 

io49 

1062 

26 

33 

5543 

5556 

5568 

558i 

5593 

56o5 

26 

34 

io75 

1088 

IIOI 

in4 

II27 

1139 

25 

34 

56i8 

563o 

5642 

5655 

5667 

5679 

25 

35 

Il52 

n65 

n78 

n9i 

I2O4 

I2I7 

24 

35 

5692 

5704 

57i7 

5729 

574i 

5754 

a4 

36 

1229 

1242 

1255 

1268 

I28l 

1294 

23 

36 

5766 

5778 

579i 

58o3 

58i5 

5828 

23 

37 

i3o6 

i3i9 

i332 

1  345 

i358 

i37i 

22 

3? 

584o 

5852 

5865 

5877 

5889 

59os 

22 

38 

i383 

i396 

i4o9 

1422 

i435li448 

21 

38 

5914 

5926 

5939 

595i 

5963 

5976 

21 

39 

i46o 

i473 

i486 

i499 

i5i2  i525 

20 

39 

5988 

6000 

6oi3 

6025 

6o37 

6o5o 

2O 

4o 

9.93i537 

f55o 

i563 

1576 

i589  1601 

19 

4o 

9.936o62 

6074 

6o87 

6o99 

6111 

6124 

19 

4i 

1614 

1627 

1640 

i653 

1666 

i678 

18 

4i 

6i36 

6:48 

6161 

6i73 

6i85 

6i98 

18 

42 

i69i 

i7o4 

I7i7 

1730 

I742 

i755 

17 

42 

6210 

6222 

6234 

6247 

6259 

627I 

17 

43 

i768 

1781 

i794 

1806 

1819  i832 

16 

43 

6284 

6296 

63o8 

6320 

6333 

6345 

16 

44 

i845 

i857 

i87o 

i883 

1896  1909 

i5 

44 

6357 

6370 

6382 

6394 

64o6 

64i9 

i5 

45 

I92I 

i934 

i947 

i96o 

T972 

i985 

i4 

45 

643i 

6443 

6456 

6468 

648o 

649s 

i4 

46 

i998 

2OII 

2024 

2o36 

2049  2062 

i3 

46 

65o5 

65i7 

6529 

6542 

6554 

6566 

i3 

47 

2O75 

2087 

2IOO 

2Il3 

2126  2i38 

12 

4? 

6578 

659i 

66o3 

66i5 

6627 

664o 

12 

48 

2l5l 

2164 

2177 

2189 

22O2  22l5 

II 

48 

6652 

6664 

6676 

6689 

67oi 

67i3 

49 

2228 

2240 

2253 

2266 

227g  2291 

IO 

49 

6725 

6738 

675o 

6762 

6774 

6787 

IO 

5o 

9.9323o4 

23l7 

2329 

2342 

23552368 

9 

5o 

o.936799 

6811 

6823 

6836 

6848 

6860 

9 

5i 

238o 

2393 

2406 

2419 

243l 

2444 

8 

5i 

6872 

6884 

6897 

69o9 

692I 

6933 

8 

52 

2457 

2469  2482 

2495 

25o8  252O 

7 

52 

6946 

6958 

697o 

6082 

6994 

7oo7 

7 

53 

2533 

2546, 

2558 

257i 

2584  269-7 

6 

53 

7oi9 

7o3i 

7o43 

7o56 

7o68 

7o8o 

6 

54 

2609 

2622 

2635 

2647 

2660  2673 

5 

54 

7o92 

7104 

7117 

7I29 

7i4i 

7i53 

5 

55 

2685 

260.8 

2711 

2724 

2736  2749 

4 

55 

7i65 

7178 

7190 

72O2 

72l4 

7226 

4 

56 

2762 

2774 

2787 

2800 

2812  2825 

3 

56 

7238 

725l 

7263 

7275 

7287 

7299 

3 

57 

2838 

285o! 

2863 

2876 

2888  2901 

2 

57 

73l2 

7324 

7336 

7348 

736o 

7372 

2 

58 

2914 

2926 

2939 

2952 

2964  2977 

I 

58 

7385 

7397 

7409 

742I 

7433 

7446 

I 

59 

2990 

3OO2 

3oi5 

3o283o4o3o53 

O 

59 

7458 

747o 

7482 

7494  75o6i75i8  o 

GO" 

50" 

40"  |  30"  1  20"   10" 

d 

60"     50"   40"  |  30"  |  20"   10" 

Co-sine  of  31  Degrees. 

g 

Co-sine  of  30  Degrees. 

p  p  A  I"  2"  3"  4"  5"  6"  7"  8"  9" 
{  1   3   4   5   6   8   9  10  12 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 
P.  Fart  J  !   2   4   5   6   7   9  10  11 

LOGARITHMIC    TANGENTS. 


i 

Tangent  of  58  Degrees. 

.« 

Tangent  of  59  Degrees. 

§ 

0" 

10"   20" 

30" 

40"   50" 

3 

0" 

10"   20"   30" 

40" 

50" 

0 

10.20421  i 

4258 

43o4 

435i 

4398 

4445 

59 

0 

10.221226 

I274 

l322 

i36g 

i4i7 

i465 

59 

I 

4492 

4539 

4586 

4633 

4679 

4726 

58 

i 

l5l2 

i56o 

l6o8 

i656 

I7o3 

1751 

58 

2 

47?3 

4820 

4867 

49i4 

496i 

5oo8 

57 

2 

i799 

i846 

i894 

1942 

1990 

2037 

57 

3 

5o54 

5ioi 

5i48 

5i95 

5242 

5289 

56 

3 

2o85 

2i33 

2181 

2228 

2276 

2324 

56 

4 

5336 

5383 

543o 

5477 

5524 

557o 

55 

4 

2372 

24i9 

2467 

25i5 

2563 

26lO 

55 

5 

5617 

5664 

5711 

5758 

58o5 

5852 

54 

5 

2658 

27o6 

2754 

2801 

2849 

2897 

54 

6 

5899 

5946 

5993 

6o4o 

608  7 

6i34 

53 

6 

2945 

2993 

3o4o 

3o88 

3i36 

3i84 

53 

7 

6181 

6227 

6274 

632i 

6368 

64i5 

52 

7 

3232 

3279 

3327 

3375 

3423 

347i 

r 

8 

6462 

65o9 

6556 

66o3 

665o 

6697 

5i 

8 

35i8 

3566 

36i4 

3662 

37io 

3757 

5i 

9 

6744 

679i 

6838 

6885 

6932 

6979 

5o 

9 

38o5 

3853 

39oi 

3949 

3997 

4o44 

5o 

10 

10.207026 

7o73 

7120 

7167 

72l4 

726l 

49 

10 

IO.224O92 

4i4o 

4i88 

4236 

4284 

4332 

49 

ii 

73o8 

7355 

7402 

7449 

7496 

7543 

48 

ii 

4379 

442  7 

4475 

4523 

457i 

46i9 

48 

12 

7590 

7637 

7684 

773i 

7778 

7825 

47 

12 

4667 

47i4 

4762 

48io 

4858 

49o6 

47 

i3 

7872 

79i9 

7966 

8oi3 

8060 

8io7 

46 

i3 

4954 

5OO2 

5o5o 

5o98 

5i45 

5i93 

46 

i4 

8!54 

8201 

8248 

8295 

8342 

8389 

45 

i4 

524i 

5289 

5337 

5385 

5433 

548! 

45 

i5 

8437 

8484 

853i 

8578 

8625 

8672 

44 

i5 

5529 

5577 

5625 

5672 

5720 

5768 

44 

16 

8719 

8766 

88i3 

8860 

89o7 

8954 

43 

16 

58i6 

5864 

59I2 

5960 

6008 

6o56 

43 

*7 

9001 

9o48 

9095 

9i43 

9i9o 

9237 

42 

17 

6io4 

6i52 

6200 

6248 

6296 

6344 

42 

18 

9284 

933i 

9378 

9425 

9472 

95l9 

4i 

18 

6392 

644o 

6488 

6535 

6583 

663i 

4i 

19 

9566 

9614 

9661 

97o8 

9755 

9802 

4o 

T9 

6679 

6727 

6775 

6823 

687i 

6919 

4o 

20 

10.209849 

9896 

9943 

999  i 

..38 

..85 

39 

20 

IO.226967 

7oi5 

7o63 

7111 

7i59 

7207 

39 

21 

IO.2IOI32 

0179 

0226 

0273 

0321 

o368 

38 

21 

?255 

73o3 

735i 

7399 

7447 

7495 

38 

22 

o4i5 

0462 

0509 

o556 

o6o3 

o65i 

37 

22 

7543 

759i 

7639 

7688 

7736 

7784 

37 

23 

0698 

o745 

0792 

o839 

0886 

o934 

36 

23 

7832 

788o 

7928 

7976 

8024 

8o72 

36 

24 

0981 

1028 

1075 

1122 

1170 

1217 

35 

24 

8120 

8168 

8216 

8264 

83i2 

836o 

35 

25 

1264 

i3n 

i358 

i4o5 

i453 

i5oo 

34 

25 

84o8 

8456 

85o4 

8552 

8601 

8649 

34 

26 

i547 

i594 

i64i 

i689 

1736 

1783 

33 

26 

8697 

8745 

8793 

884i 

8889 

8937 

33 

27 

i83o 

1878 

1925 

I972 

2OI9 

2066 

32 

27 

8985 

9o33 

9o8i 

9i3o 

9i78 

9226 

32 

28 

2Il4 

2161 

2208 

2255 

23o3 

235o 

3i 

28 

9274 

9322 

937o 

94i8 

9466 

95i5 

3i 

29 

2397 

2444 

2492 

2539 

2586 

2633 

3o 

29 

9563 

96n 

9659 

9707 

9755 

9803 

3o 

3o 

10.212681 

2728 

2775 

2822 

2870 

2917 

29 

3o 

I0.229852 

99oo 

9948 

9996 

..44 

..92 

29 

3i 

2964 

3oia 

3o59 

3io6 

3i53 

3201 

28 

3i 

io.23oi4o 

oi89 

0237 

0285 

o333 

o38i 

28 

32 

3248 

3295 

3343 

339o 

3437 

3484 

27 

32 

0429 

o478 

o526 

o574 

0622 

o67o 

27 

33 

3532 

3579 

3626 

3674 

3721 

3768 

26 

33 

o7i9 

o767 

o8i5 

o863 

o9ii 

0960 

26 

34 

38i6 

3863 

3910 

3958 

4oo5 

4o52 

25 

34 

1008 

io56 

no4 

1.  1  52 

I2OI 

1249 

25 

35 

4ioo 

4i47 

4i94 

4242 

4289 

4336 

24 

35 

I297 

1  345 

i394 

1442 

i49o 

i538 

24 

36 

4384 

443  1 

4478 

4526 

4573 

4620 

23 

36 

i586 

i635 

i683 

1731 

i779 

1828 

23 

37 

4668 

47'5 

4762 

48io 

4857 

4905 

22 

37 

i876 

I924 

I973 

2O2I 

2o69 

2II7 

22 

38 

4952 

4999 

5o47 

5o94 

5i4i 

5189 

21 

38 

2166 

22l4 

2262 

23lO 

2359 

2407 

21 

39 

5236 

5284 

533i 

5378 

5426 

5473 

20 

39 

2455 

25o4 

2552 

26OO 

2648 

2697 

20 

4o 

IO.2I552I 

5568 

56i5 

5663 

5710 

5758 

T9 

4o 

I0.232745 

2793 

2842 

2890 

2938 

2987 

J9 

4i 

58o5 

5852 

5900 

5947 

5995 

6042 

18 

4i 

3o35 

3o83 

3i32 

3i8o 

3228 

3277 

18 

42 

6o9o 

6!37 

6i84 

6232 

6279 

6327 

*7 

4s 

3325 

3373 

3422 

3470 

35i8 

3567 

ll 

43 

6374 

6422 

6469 

65i7 

6564 

6612 

16 

43 

36i5 

3663 

37I2 

3760 

38o8 

3857 

16 

44 

6659 

6706 

6754 

6801 

6849 

6896 

i5 

44 

39o5 

3953 

4OO2 

4o5o 

4o99 

4i47 

i5 

45 

6944 

699I 

7o39 

7o86 

7i34 

7181 

i4 

45 

4i95 

4244 

4292 

434o 

4389 

4437 

i4 

46 

7229 

7276 

7324 

73?I 

74i9 

7466 

i3 

46 

4486 

4534 

4582 

463i 

4679 

4728 

i3 

4? 

7614 

756i 

76°9 

7656 

77°4 

775i 

12 

47 

4776 

4825 

4873 

4921 

4970 

5oi8 

12 

48 

7799 

7846 

7894 

794i 

7989 

8o36 

I  I 

48 

5o67 

5n5 

5i64 

5212 

526o 

53o9 

II 

49 

8084 

8i3i 

8i79 

8226 

8274 

8322 

10 

49 

5357 

54o6 

5454 

55o3 

555i 

56oo 

IO 

5o 

io.2i8369 

8417 

8464 

85i2 

8559 

86o7 

9 

5o 

io.235648 

5696 

5745 

5793 

5842 

589o 

9 

5i 

8654 

8702 

875o 

8797 

8845 

8892 

8 

5i 

5939 

5987 

6o36 

6o84 

6i33 

6181 

8 

52 

894o 

8987 

9o35 

9o83 

9i3o 

9178 

7 

52 

623o 

6278 

6327 

6375 

6424 

6472 

7 

53 

9225 

9273 

932I 

9368 

94i6 

9463 

6 

53 

652i 

6569 

6618 

6666 

67i5 

6763 

6 

54 

95n 

9559 

96o6 

9654 

97oi 

9749 

5 

54 

6812 

6860 

6909 

6957 

7006 

7o55 

5 

55 

9797 

9844 

9892 

9939 

9987 

..35 

4 

55 

7io3 

7l52 

720O 

7249 

7297 

7346 

4 

50 

10.220082 

oi3o 

oi78 

0225 

0273 

0321 

3 

56 

7394 

7443 

7492 

754o 

7589 

7637 

3 

5? 

o368 

o4i6 

o463 

o5n 

o559 

0606 

2 

57 

7686 

7734 

7783 

7832 

788o 

7929 

2 

58 

o654 

070-?. 

o749 

°797 

o845 

0892 

I 

58 

7977 

8026 

8o75 

8i23 

8172 

8220 

I 

59 

o94o 

o988|io36 

io83 

n3i 

1179 

O 

59 

826983i88366 

84i5 

8463 

85i2 

O 

GO" 

50"   40" 

30" 

20" 

10" 

S" 

60"    1  50"   40"   30"   20"   10" 

Co-tangent  of  31  Degrees. 

3 

Co-tangent  of  30  Degrees. 

ii 

1  P  Part  5  l"  ~"  3"  4"  5"  6"  7//  8//  9" 
}  5   9   14  19  24  28  33  38  43 

p  T)  .  S  1"  2"  3"  4"  5"  6"  7"  8"  9" 
111  1  5   10f  14  19  24  20  34  39  43 

LOGARITHMIC    SINES. 


c 

Sine  of  60  Degrees.      | 

1  = 

Sine  of  61  Deofrees. 

1 

§ 

0" 

10"  '  -20" 

30" 

40" 

50" 

si 

0" 

10" 

20"  |  30"  |  40" 

50" 

o 

9.93753i 

7543 

7555 

7567 

7579 

7591 

59 

o 

9.94*819 

i83i 

1843 

i854 

1866 

1878 

59 

I 

7602 

76i6 

7628 

764o 

7652 

7664 

58 

I 

1889 

I9OI 

I9i3 

1924 

i936 

i948 

53 

2 

7676 

7689 

77oi 

77i3 

7725 

7737 

57 

2 

i959 

1971 

1983 

1994 

2006 

2017 

07 

3 

7749 

7761 

7773 

7786 

7798 

7810 

56 

3 

2029 

2O52 

2064 

2076 

2087 

56 

4 

•7822 

7834 

7846 

7858 

7870 

7883 

55 

4 

2099 

2III 

2122 

2134 

2146 

2l57 

55 

5 

7895 

7907 

79i9 

793i 

7943 

7955 

54 

5 

2169 

2l8o 

2192 

22O4 

22l5 

2227 

54 

6 

7967 

7979 

7992 

8oo4 

8016 

8028 

53 

6 

2239 

2250 

2262 

2273 

2285 

2297 

53 

7 

8o4o 

8o52 

8o64 

8076 

8088 

8lOO 

52 

7 

23o8 

2320 

233l 

2343 

2355 

2366 

5a 

8 

8n3 

8i25 

8i37 

8i49 

8161 

8i73 

5i 

8 

2378 

2390 

2401 

24i3 

2424 

2436 

5i 

9 

8i85 

8197 

8209 

8221 

8233 

8245 

5o 

9 

2448 

2459 

2471 

2482 

s494 

25o6 

5o 

10 

9.  938258 

8270 

8282 

8294 

83o6 

83i8 

49 

10 

9.9425i7 

2529 

254o 

2552 

2563 

2575 

49 

ii 

833o 

8342 

8354 

8366 

8378 

839o 

48 

ii 

2587 

2598 

2610 

2621 

2633 

2645 

48 

12 

8402 

84i4 

8426 

8439 

845  1 

8463 

47 

12 

2656 

2668 

2679 

2691 

2702 

2714 

47 

i3 

8475 

8487 

8499 

85n 

8523 

8535 

46 

i3 

2726 

2737 

2749 

2760 

2772 

2783 

46 

i4 

8547 

8559 

857i 

8583 

8595 

86o7 

45 

i4 

2795 

2806 

2818 

283o 

2841 

2853 

45 

i5 

8619 

863i 

8643 

8655 

8667 

8679 

44 

i5 

2864 

2876 

2887 

2899 

2910 

2922 

44 

16 

8691 

87o3 

87i5 

8727 

8739 

875i 

43 

16 

2934 

2945 

2957 

2968 

2980 

299I 

43 

17 

8763 

8776 

8788  8800 

8812 

8824 

42 

i7 

3oo3 

3oi4 

3026 

3o37 

3o49 

3o6o 

42 

18 

8836 

8848 

88608872 

8884 

8896 

4i 

18 

3072 

3o83 

3o95 

3107 

3n8 

3i3o 

4i 

i9 

8908 

8920 

8932  8944 

8956 

8968 

4o 

I9 

3i4i 

3i53 

3i64 

3i76 

3i87 

3i99 

4o 

20 

9.938980 

8992 

90049016 

9028 

9040 

39 

20 

9.9432io 

3222 

3233 

3245 

3256 

3268 

39 

21 

9o52 

9064 

9076  9087 

9°99 

9111 

38 

21 

3279 

329I 

3302 

33i4 

3325 

3337 

38 

22 

9123 

9i35 

9147  9i59 

9171 

9i83 

37 

22 

3348 

336o 

3371 

3383 

3394 

34o6 

37 

23 

9195 

9207 

92I9  923l 

9243 

9255 

36 

23 

3417 

3429 

344o 

3452 

3463 

3475 

36 

24 

9267 

9279 

9291^303 

93i5 

9327 

35 

24 

3486 

3498 

3509 

352i 

3532 

3543 

35 

25 

9339 

93639375 

9387 

9399 

34 

25 

3555 

3566 

3578 

3589 

36oi 

36i2 

34 

26 

9422 

9434  9446 

9458 

33 

26 

3624 

3635 

3647 

3658 

367o 

368i 

33 

27 

9482 

9494 

95069518 

953o 

9542 

32 

27 

3693 

3704 

37i5 

3727 

3738 

375o 

32 

28 

9554 

9566 

9578  9590 

96oi 

96i3 

3i 

28 

376i 

3773 

3784 

3796 

38o7 

38i8 

3: 

29 

9625 

9637 

9649  9661 

9673 

9685 

3o 

29 

383o 

384i 

3853 

3864 

3876 

3887 

3o 

30:9.989697 

9709 

9721  9733 

9744 

9756 

29 

3o 

9.943899 

3910 

3921 

3g33 

3944 

3956 

29 

3i 

9768 

9780 

9792  9804 

9816 

9828 

28 

3i 

3967 

3978 

399o 

4ooi 

4oi3 

4024 

28 

32 

9840 

9852 

9863:9875 

9887 

9899 

27 

32 

4o36 

4047 

4o58 

4070 

4o8i 

4o93 

27 

33 

9911 

9923 

9935  9947 

9959 

9970 

26 

33 

4io4 

4n5 

4127 

4i38 

4i5o 

4i6i 

26 

34 

9982 

9994 

...6  ..18 

..3o 

..42 

25 

34 

4172 

4x84 

4i95 

4207 

4218 

4229 

25 

35 

9.940054 

oo65 

oo77  0089 

OIOI 

on3 

24 

35 

4241 

4252 

4264 

4275 

4286 

4298 

24 

36 

0125 

0/87 

oi48  0160 

0172 

oi84 

23 

36 

4309 

4321 

4332 

4343 

4355 

4366 

23 

37 

0196 

0208 

O22O  O23l 

0243 

0255 

22 

37 

4377 

4389 

44oo 

44i2 

4423 

4434 

22 

38 

0267 

0279 

0291  o3o3 

o3i4 

o326 

21 

38 

4446 

4457 

4468 

448o 

449i 

45o3 

21 

39 

o338 

o35o 

o362  o374 

o385 

°397 

20 

39 

45i4 

4525 

4537 

4548 

4559 

457i 

20 

4o 

9.940409 

0421 

o433  o445 

o456 

o468 

I9 

4o 

9.944582 

4593 

46o5 

4616 

4627 

4639 

19 

4i 

o48o 

0492 

o5o4  o5i6 

0527 

o539 

18 

4i 

465o 

466i 

4673 

4684 

4696 

4707 

18 

42 

o55i 

o563 

o575j0586 

o598 

0610 

i7 

42 

4718 

4780 

4741 

4752 

4764 

4775 

I7 

43 

0622 

o634 

o645o657 

0669 

0681 

16 

43 

4786 

4798 

4809 

4820 

483i 

4843 

16 

44 

0693 

0704 

0716,0728 

0740 

0752 

i5 

44 

4854 

4865 

4877 

4888 

4899 

4911 

i5 

45 

0763 

o775 

07870799 

0811 

0822 

i4 

45 

4922 

4933 

4945 

4956 

4967 

4979 

i4 

46 

o834 

o846 

o858 

0870 

0881 

0893 

i3 

46 

4990 

5ooi 

5oi3 

5o24 

5o35 

5o46 

i3 

47 

0905 

o9i7 

0928 

o94o 

0952 

0964 

12 

4? 

5o58 

5069 

5o8o 

5092 

5io3 

5n4 

12 

48 

0975 

o987 

0999 

IOII 

1023 

io34 

II 

48 

5i25 

5i37 

5i48 

5i5g 

5171 

5i82 

II 

49 

1046 

io58 

1070 

1081 

1093 

no5 

10 

49 

5i93 

52i6 

5227 

5238 

525o 

10 

5o 

9.941117 

1128 

n4o 

Il52 

1164 

1175 

9 

5o 

9.945261 

5272 

5283 

5295 

53o6 

53i7 

9 

5i 

1187 

1199 

I2II 

1222 

1234 

1246 

8 

5i 

5328 

534o 

535i 

5362 

5374 

5385 

8 

52 

1258 

1269 

I28l 

1293 

i3o4 

i3i6 

7 

52 

5396 

5407 

5419 

543o 

544i 

5452 

7 

53 

1328 

i34o 

i35i 

i363 

i375 

i387 

6 

53 

5464 

5475 

5486 

5497 

55o9 

5520 

6 

54 

1398 

i4io 

1422 

i433 

1  445 

i457 

5 

54 

553i 

5542 

5554 

5565 

5576 

5587 

5 

55 

1469 

i48o 

1492 

i5o4 

i5i5 

1527 

4 

55 

5598 

56io 

562i 

5632 

5643 

5655 

4 

56 

i539 

i55o 

i562 

i574 

i586 

l597 

3 

56 

5666 

5677 

5688 

5700 

57n 

5722 

3 

57 

1609 

1621 

i632 

1  644 

i656 

1667 

2 

57 

5733 

5744 

5756 

5767 

5778 

5789 

2 

58 

1679 

i69i 

1702 

1714 

1726 

i738 

I 

58 

58oo 

58i2  5823 

5834  5845 

5857 

I 

59 

1749 

1761 

i773 

1784 

i796 

1808 

O 

59 

5868 

5879  5890 

5901  5gi3 

5924 

0 

60" 

50" 

40" 

30"   20" 

10" 

a 

60" 

50"   40" 

30"   20"   10" 

j 

Co-sine  of  29  Degrees. 

3 

Co-sine  of  28  Degrees. 

1 

P  TartJ  l"  2"  3"  4"  5"  6"  7"  8"  9" 

^  i"  2"  1"  4"  5"  6"  7"  8"  9" 
P.Part^  !   2   3   5   6   7   8   9  10 

LOGARITHMIC    TANGENTS. 


.5 

Tangent  of  60  Degrees. 

A 

Tangent  of  61  Degrees. 

3 

0"    |  10" 

20" 

30" 

40" 

50" 

al 

0" 

10" 

20" 

30"! 

40" 

50" 

o 

io.23856i 

8609 

8658 

8707 

8755 

88o4 

59 

0 

10.266248 

6298 

6347 

6397 

6447 

6496 

59 

i 

8852 

8901 

895o 

8998 

9047 

9096 

58 

I 

6546 

6696 

6645 

6696 

6745 

6794 

58 

2 

9144 

9!93 

9242 

9290 

9339 

9388 

57 

2 

6844 

6894 

6944 

6993 

7043 

7093 

57 

3 

9436 

9485 

9534 

9582 

963i 

9680 

56 

3 

7142 

7192 

7242 

7291 

7341 

739i 

56 

4 

9728 

9777 

9826 

9874 

9923 

9972 

55 

4 

744i 

7490 

754o 

7690 

7640 

7689 

55 

5 

10.240021 

0069 

0118 

0167 

O2l5 

0264 

54 

5 

7739 

7789 

7839 

7888 

7938 

7988 

54 

6 

o3i3 

o362 

o4io 

o459 

o5o8 

o557 

53 

6 

8o38 

8087 

8i37 

8187 

8237 

8286 

53 

7 

o6o5 

o654 

0703 

0762 

0800 

0849 

52 

7 

8336 

8386 

8436 

8486 

8535 

8585 

D2 

8 

0898 

0947 

0995 

io44 

io93 

1142 

5i 

8 

8635 

8685 

8735 

8784 

8834 

8884 

5i 

9 

1190 

1239 

1288 

i337 

i385 

i434 

5o 

9 

8934 

8984 

9033 

9083 

9i33 

9i83 

5o 

10 

io.24i483 

i532 

1681 

i629 

1678 

1727 

49 

10 

io.259233 

9283 

9332 

9382 

9432 

9482 

49 

ii 

1776 

i825 

1873 

I922 

1971 

2O20 

48 

ii 

9532 

9682 

9632 

968i 

973i 

978i 

48 

12 

2069 

2118 

2166 

22l5 

2264 

23l3 

47 

12 

983i 

9881 

993i 

998i 

..3i 

..80 

47 

i3 

2362 

2411 

2459 

2608 

2557 

2606 

46 

i3 

io.26oi3o 

0180 

O23O 

0280 

o33o 

o38o 

46 

i4 

2655 

2704 

2753 

2801 

2860 

2899 

45 

i4 

o43o 

o48o 

o53o 

0680 

o629 

0679 

45 

i5 

2948 

2997 

3o46 

3o95 

3i43 

3192 

44 

i5 

0729 

0779 

o829 

0879 

0929 

0979 

44 

16 

324i 

3290 

3339 

3388 

3437 

3486 

43 

16 

I029 

1079 

II29 

1179 

I229 

i279 

43 

17 

3535 

3584 

3632 

368i 

373o 

3779 

42 

J7 

i329 

i379 

i429 

1479 

l529 

i579 

42 

18 

3828 

3877 

3926 

3975 

4024 

4o73 

4i 

18 

i629 

i679 

I729 

1779 

i829 

i879 

4i 

'9 

4l22 

4171 

4220 

4269 

43i8 

4366 

4o 

T9 

1929 

i979 

2O29 

2079 

2I29 

2I79 

4o 

20 

i  O.2444  i  5 

4464 

45i3 

4§62 

46n 

466o 

39 

20 

10.262229 

2279 

2329 

2379 

2429 

2479 

39 

21 

4709 

4758 

48o7 

4856 

4go5 

4954 

38 

21 

2629 

2579 

2629 

2679 

2729 

2779 

38 

22 

5oo3 

5o52 

5ioi 

5i5o 

5l99 

5248 

37 

22 

2829 

2879 

2929 

2979 

3o29 

3o79 

37 

23 

5297 

5346 

53g5 

5444 

5493 

5542 

36 

23 

3i3o 

3i8o 

3230 

328o 

333o 

338o 

36 

24 

559i 

564o 

5689 

5738 

5787 

5836 

35 

24 

343o 

348o 

353o 

358o 

363o 

368i 

35 

25 

5885 

5934 

698-4 

6o33 

6082 

6i3i 

34 

25 

3731 

378i 

383i 

388i 

393i 

398i 

34 

26 

6180 

6229 

62-78 

6327 

6376 

6425 

33 

26 

4o3i 

4082 

4l32 

4182 

4232 

4282 

33 

27 

6474 

6523 

6672 

6621 

6670 

6720 

32 

27 

4332 

4382 

4433 

4483 

4533 

4583 

32 

28 

6769 

6818 

6867 

6916 

6965 

7014 

3i 

28 

4633 

4683 

4734 

4784 

4834 

4884 

3i 

29 

7063 

7112 

7161 

72II 

7260 

73o9 

3o 

29 

4934 

4985 

5o35 

6086 

5i35 

5i85 

3o 

3o 

io.247358 

7407 

7456 

75o5 

7554 

7604 

29 

3o 

10.266236 

6286 

5336 

5386 

5436 

5487 

29 

3i 

7653 

7702 

775i 

78oo 

7849 

7899 

28 

3i 

5537 

5587 

5637 

5688 

5738 

6788 

28 

32 

7948 

7997 

8o46 

8o95 

8i44 

8194 

27 

32 

5838 

5889 

5939 

6989 

6o39 

6o9o 

27 

33 

8243 

8292 

834i 

839o 

8439 

8489 

26 

33 

6i4o 

6i9o 

6240 

6291 

634i 

639i 

26 

34 

8538 

8587 

8636 

8685 

8735 

8784 

25 

34 

6442 

6492 

6542 

6692 

6643 

6693 

26 

35 

8833 

8882 

893i 

8981 

9o3o 

9079 

24 

35 

6743 

6794 

6844 

6894 

6945 

6996 

24 

36 

9128 

9178 

9227 

9276 

9325 

9375 

23 

36 

7045 

7o96 

7i46 

7196 

7247 

7297 

23 

37 

9424 

9473 

9522 

9572 

962I 

967o 

22 

3? 

7347 

7398 

7448 

7498 

7549 

7699 

22 

38 

9719 

9769 

9818 

9867 

99i6 

9966 

21 

38 

7649 

77oo 

7760 

7800 

7861 

7901 

21 

39 

io.25ooi5 

oo64 

on4 

oi63 

O2I2 

0261 

20 

39 

7962 

8002 

8062 

8io3 

8i53 

8204 

2O 

4o 

io.25o3n 

o36o 

0409 

0459 

o5o8 

o557 

19 

4o 

10.268264 

83o4 

8355 

84o5 

8456 

85o6 

J9 

4i 

0607 

o656 

0706 

o755 

o8o4 

o853 

18 

4i 

8556 

86o7 

8667 

8708 

8768 

8809 

18 

42 

ogoS 

0952 

IOOI 

io5i 

IIOO 

n49 

J7 

42 

8869 

89o9 

896o 

9010 

9o6i 

9111 

J7 

43 

1199 

1248 

1297 

1  347 

i396 

1  445 

16 

43 

9162 

92I2 

9263 

93i3 

9364 

94i4 

16 

44 

i495 

1  544 

i594 

1  643 

l692 

1742 

16 

44 

9465 

95i5 

9566 

9616 

9667 

9717 

i5 

45 

1791 

i84o 

1890 

i939 

i989 

2038 

i4 

45 

9767 

9818 

9868 

9919 

9970 

.  .20 

i4 

46 

2087 

2137 

2186 

2236 

2285 

2335 

i3 

46 

10.270071 

0121 

0172 

O222 

0273 

o323 

i3 

4? 

2384 

2433 

2483 

2532 

2682 

263i 

12 

47 

o374 

0424 

0476 

0525 

0676 

0626 

12 

48 

2681 

2730 

2779 

2829 

2878 

2928 

II 

48 

0677 

0728 

0778 

0829 

0879 

ogSo 

II 

49 

2977 

3027 

3076 

3i26 

3i75 

3225 

10 

49 

0980 

io3i 

1082 

Il32 

i*83 

1233 

IO 

5o 

10.253274 

3324 

3373 

3423 

3472 

352i 

g 

5o 

10.271284 

i335 

i385 

i436 

i486 

i537 

9 

5i 

357i 

3620 

367o 

37i9 

3769 

38i8 

8 

Si 

1  588 

i638 

i689 

i739 

1790 

i84i 

8 

52 

3868 

3918 

3967 

4017 

4o66 

4n6 

7 

52 

1891 

1942 

i993 

2o43 

2094 

2i45 

7 

53 

4i65 

42i5 

4264 

43i4 

4363 

44i3 

6 

53 

2196 

2246 

2297 

2347 

2398 

2449 

6 

54 

4462 

45i2 

456i 

46n 

466i 

4710 

5 

54 

2499 

2660 

26OI 

2661 

2702 

276  3 

5 

55 

476o 

4809 

4859 

4908 

4958 

5oo8 

4 

55 

28o3 

2864 

29o5 

2955 

3oo6 

3o57 

4 

56 

5o57 

5io7 

5i56 

52o6 

5256 

53o5 

3 

56 

3io8 

3i58 

3209 

3260 

33io 

336i 

3 

57 

5355 

54o4 

5454 

55o4 

5553 

56»3 

2 

57 

34i2 

3463 

35i3 

3564 

36i5 

3666 

2 

58 

5652 

5702 

6762 

58oi 

585i 

59oi 

I 

58 

37i6 

3767 

38i8 

3869 

39i9 

397o 

I 

59 

595o 

6000 

6049 

6099 

6i49 

6i98 

0 

59 

4021^072 

4l22 

4i?3 

4224 

4276 

0 

60" 

50" 

40" 

30" 

20" 

10" 

c 

60"     |  50" 

40" 

30" 

20"  1  10" 

rj 

Co-tangent  of  29  Degrees. 

2 

Co-tangent  of  28  Degrees. 

1 

P  PnrtJ  l"  2"  3"  4"  3"  G"  7//  8"  9" 

p  p   (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

{  5  10  15  20  25  29  34  39  44 

f  5  10  15  20  25  3<)  35  40  45 

LOGARITHMIC    SINES. 


.9 

Sine  of  62  Degrees 

C 

Sine  of  63  Degrees. 

i 

0" 

10" 

20"   30"   40" 

50" 

& 

0" 

10" 

20" 

30"  ]  40" 

5(1" 

o 

I 

9.945935 
6002 

5946 
6oi3 

59575969598o 

6o24;6o36  6o47 

599i 
6o58 

59 

58 

o 
I 

9.949881 
9945 

9892 
9956 

9902 
9967 

99i3 
9977 

9924 
9988 

9935 
9999 

59 

58 

2 

6069 

60806092  6io3  6n4 

6i25 

57 

2 

9.950010 

O02O 

oo3i 

0042 

oo52 

oo63 

5? 

3 

6i36 

6i47  6159  6170 

6181 

6192 

56 

3 

oo74 

0084 

0095 

0106 

011-7 

012-7 

56 

4 

62o3 

621462266237 

6248 

6259 

55 

4 

oi38 

0149 

oi59 

01-70 

0181 

0191 

55 

5 

6270 

6281  6293,6304 

63i5 

6326 

54 

5 

O2O2 

0213 

0224 

0234 

0245 

0256 

54 

6 

6337 

6348  6359  6371 

6382 

6393 

53 

6 

O266 

0277 

0288 

0298 

oSog 

0320 

53 

7 

8 

64o4 
6471 

64i564a6643? 

648264g365o4 

6449 
65i5 

646o 
6526 

52 

5i 

8 

o33o 
o394 

o34i 
o4o5 

o352 
o4i6 

O362 

0426 

o373 
o437 

o384 
o448 

52 

5i 

9 

6538 

6549 

656o  657i 

6582 

6593 

5o 

9 

o458 

0469 

o48o 

0490 

o5oi 

O5l2 

5o 

10 

9.946604 

66i5 

66276638 

6649 

6660 

49 

IO 

9.95o522 

o533 

o544 

o554 

o565 

o576 

49 

ii 

6671 

6682 

6693  67o4 

67i5 

6-726 

48 

ii 

o586 

0597 

0607 

0618 

0629 

0639 

48 

12 

6738 

6749  6760  6771 

6782 

6793 

47 

12 

o65o 

0661 

0671 

0682 

0693 

o7o3 

47 

i3 

i4 

68o4 
6871 

681  5  6826  6837 
6883  6893  6904 

6849 
69i5 

6860 
6926 

46 

45 

i3 

i4 

o7i4 
0778 

0724 
0788 

o735 
°799 

o746 
0809 

o756 
0820 

o767 
o83i 

46 

45 

i5 

6937 

6948 

6959 

697o 

6982 

6993 

44 

i5 

o84i 

o852 

0862 

o873 

08  84 

0894 

44 

16 

7004 

70i5 

7O26 

7o37 

7048 

7o59 

43 

16 

o9o5 

ogi5 

0926 

o937 

o947 

o958 

43 

17 

7070 

7081 

•7092 

7io3 

7114 

7I25 

42 

17 

o968 

0979 

0990 

IOOO 

IOII 

IO2I 

42 

18 

7i36 

7147 

7i58 

7i7o 

7181 

7192 

4i 

18 

1032 

io43 

io53 

io64 

io74 

io85 

4i 

19 

7203 

7214  7225 

7236 

7247 

7258 

4o 

19 

io96 

1106 

1117 

I  I27 

n38 

n48 

4o 

20 

9-947269 

729I 

7302 

73i3 

7324 

39 

20 

9.951159 

1170 

1180 

II9I 

I2OI 

1212 

39 

21 

7335 

7346  7357 

7368 

7379 

739o 

38 

21 

1222 

1233 

1244 

1254 

1265 

I275 

38 

22 

7401 

74l2 

7423 

7434 

7445 

7456 

37 

22 

1286 

1296 

1307 

i3i7 

i328 

i339 

37 

23 

7467 

7478;7489 

75oo 

75n 

7522 

36 

23 

i349 

i36o 

1370 

i38i 

1391 

1402 

36 

24 

7533 

7545  7556 

7567 

7578 

7589 

35 

24 

1412 

i423 

i434 

i444 

i455 

i465 

35 

25 

7600 

7611 

•7622 

7633 

7644 

7655 

34 

25 

1476 

i486 

i497 

i5o7 

i5i8 

i528 

34 

26 

7665 

7676  7687 

7698 

77°9 

7720 

33 

26 

i539 

1  549 

i56o 

i57o 

i58i 

i59i 

33 

27 

773i 

7?42 

7753 

7764 

7775 

7786 

32 

27 

1602 

i6i3 

1623 

1  634 

1  644 

i655 

32 

28 

7797 

78o8  78i9 

783o 

7852 

3i 

28 

i665 

1676 

1686 

i697 

I707 

1718 

3i 

29 

7863 

7874  7885 

7896 

7907 

79i8 

3o 

29 

1728 

i739 

i749 

1760 

i77o 

1781 

3o 

3o 

9.947929 

794o  -795  1 

7962 

7973 

7984 

29 

3o 

9.951791 

1802 

1812 

1823 

i833 

1  844 

29 

3i 

7995 

8006  801-7 

8028 

8o38 

8o49 

28 

3i 

1  854 

1  865 

i875 

1886 

1896 

1907 

28 

32 

8060 

80-71 

8082 

8093 

8io4 

8n5 

27 

32 

1917 

1928 

1938 

i949 

i959 

1969 

27 

33 

8126 

8i378i48 

8i59 

8170 

8181 

26 

33 

1980 

1990 

20OI 

2OII 

2022 

2032 

26 

34 

8192 

82o382i3 

8224 

8235 

8246 

25 

34 

2043 

2o53 

20&4 

2074 

2  08  5 

2O95 

25 

35 

8257 

826882-79 

8290 

83oi 

83i2 

24 

35 

2106 

2116 

2126 

2l37 

2l47 

2i58 

24 

36 

8323 

83348344 

8355 

8366 

8377 

23 

36 

2168 

2179 

2189 

22OO 

2210 

2221 

23 

37 

8388 

83998410 

8421 

8432 

8443 

22 

37 

223l 

2241 

2252 

2262 

2273 

2283 

22 

38 

8454 

84648475 

8486 

8497 

85o8 

21 

38 

2294 

23o4 

23l4 

2325 

2335 

2346 

21 

39 

85i9 

853o  854i 

8552 

8562 

8573 

2O 

39 

2356 

2367 

2377 

2387 

2398 

2408 

20  • 

4o 

9.  948584 

859586o6 

8617 

8628 

8639 

19 

4o 

9  .952419 

2429 

a44o 

245o 

2460 

2471 

19 

4i 

865o 

8660  867i 

8682 

8693 

87o4 

18 

4i 

248  1 

2492 

25O2 

25l2 

2523 

2533 

I^  i 

42 

87i5 

87268736 

8747 

8758 

8769 

Jl 

42 

a544 

2554 

2565 

2575 

2585 

2596 

17 

43 

8780 

879i 

8802 

8812 

8823 

8834 

16 

43 

2606 

2617 

262-7 

2637 

2648 

2658 

16 

44 

8845 

88568867 

8878 

8888 

8899 

i5 

44 

2669 

2679 

2689 

27OO 

2-710 

2720 

i5 

45 

8910 

8921  8932 

8943 

8954 

8964 

i4 

45 

2731 

2741 

2752 

2762 

2772 

2783 

i4 

46 

8975 

8986 

8997 

9008 

9019 

9029 

i3 

46 

2793 

28o3 

28l4 

2824 

2835 

2845 

i3 

47 

9040 

905  1 

9062 

9o73 

9083 

9094 

12 

47 

2855 

2866 

2876 

2886 

289-7 

2907 

12 

48 

9io5 

9116 

9I27 

9i38 

9i48 

9l59 

II 

48 

2918 

2928 

2938 

2949 

2959 

2969 

II 

49 

9170 

9181 

9I92 

9202 

9213 

9224 

IO 

49 

2980 

2990 

3ooo 

3on 

3021 

3o3i 

10 

5o 
5i 

9.949235 
93oo 

9246 

9256 
932I 

9267 
9332 

9278 
9343 

9289 
9354 

I 

5o 
5i 

9<953o42 
3io4 

3o52 
3n4 

3o62 
3i24 

3o73 
3i35 

3o83 
3i45 

3i55 

9 

8 

52 

9364 

9375 

9386 

9397 

9408 

94i8 

7 

52 

3i66 

3i76 

3i86 

3i97 

3207 

3217 

7 

53 

9429 

945i 

9462 

9472 

9483 

6 

53 

3228 

3238 

3248 

3259 

3269 

3279 

6  i 

54 

9494 

95o5 

95i5 

9526 

9537 

9548 

5 

54 

3290 

33oo 

33io 

332i 

333i 

334i 

5 

55 

9558 

9569 

958o 

959J 

9602 

96l2 

4 

55 

3352 

3362 

3372 

3382 

SSgS 

34o3 

4 

56 

57 

9623 
9688 

J63| 

9698 

9645 
9-709 

9655 
9720 

9666 
973i 

9677 

974i 

3 

2 

56 

34i3 
3475 

3424 
3485 

3434 
3496 

3444 
35o6 

3455 
35i6 

3465 
3527 

3 

2 

58 

9752 

9763 

9774 

Q784 

9-795 

9806 

I 

58 

3537 

35473557  3568 

3578  3588  i 

59 

9816 

9827 

9888  9849 

9859  98-70 

O 

59 

359936093619362936403650  o 

60"     50" 

40"   30"   20"   10" 

-. 

60"     50"   40"   30"  !  20"   10"  :  c- 

Co-sine  of  27  Degrees. 

2 

Co-sine  of  26  Degrees.     ~ 

„  ,,  .(!"  2"  3"  4"  5"  6"  7"  8"  9"  II      A  I"  2"  3"  4"  5"  6"  7"  8"  9" 
F.rartJ  i   2   3   4   5   7   8   9  10   P'lart}  123456789 

LOGARITHMIC    TANGENTS. 


8-3 


4 

Tangent  of  62  Degrees. 

a 

Tangent  of  63  Degrees. 

2 

0" 

10" 

20-' 

30" 

40" 

50" 

3 

0" 

10" 

20" 

30" 

40" 

50" 

o 

10.274326 

4376 

44s7 

4478 

4529 

458o 

59 

0 

I0.292834 

2886 

2938 

299° 

3o42 

3o94 

59 

i 

463o 

468  1 

4732 

4783 

4834 

4885 

58 

i 

3i46 

3i99 

325i 

33o3 

3355 

3407 

58 

2 

4935 

4986 

5o37 

5o88 

5i39 

5190 

57 

2 

3459 

35n 

3563 

36i5 

3667 

3720 

57 

3 

5240 

5291 

5342 

5393 

5444 

5495 

56 

3 

3772 

3824 

3876 

3928 

398o 

4o32 

56 

4 

5546 

5597 

5647 

5698 

5749 

58oo 

55 

4 

4o84 

4i37 

4189 

424i 

4293 

4345 

55 

5 

585i 

5902 

5953 

6oo4 

6o55 

6io5 

54 

5 

4397 

4449 

45o2 

4554 

46o6 

4658 

54 

6 

6i56 

6207 

6258 

6309 

636o 

64n 

53 

6 

4710 

4763 

48i5 

4867 

4919 

4971 

53 

7 

6462 

65i3 

6564 

66i5 

6666 

6717 

52 

7 

5o24 

5o76 

5i28 

5i8o 

5232 

5285 

52 

8 

6768 

6819 

6870 

6920 

6971 

7022 

5i 

8 

5337 

5389 

544  1 

5494 

5546 

5598 

5i 

9 

7o73 

7124 

7175 

7226 

7277 

7328 

5o 

9 

565o 

57o3 

5755 

58o7 

5859 

59I2 

5o 

10 

10.  277379 

743o 

748  1 

7532 

7583 

7634 

49 

10 

10.295964 

6016 

6068 

6121 

6i73 

6225 

/ 

ii 

7685 

7736 

7787 

7838 

7889 

7940 

48 

ii 

6278 

633o 

6382 

6434 

6487 

6539 

13 

7991 

8o43 

8094 

8x45 

8196 

8247 

47 

12 

6591 

6644 

6696 

6748 

6801 

6853 

47 

i3 

8298 

8349 

84oo 

845  1 

85o2 

8553 

46 

i3 

6905 

6958 

7oio 

7062 

7116 

7167 

46 

i4 

86o4 

8655 

8706 

8757 

8809 

8860 

45 

i4 

7219 

7272 

7324 

7377 

7429 

748i 

45 

i5 

8911 

8962 

9013 

9064 

9n5 

9166 

44 

i5 

7534 

7586 

7638 

769i 

7743 

7796 

44 

16 

9217 

9268 

9320 

937i 

9422 

9473 

43 

16 

7848 

790o 

7953 

8oo5 

8o58 

8no 

43 

*7 

9524 

9575 

9626 

9678 

9729 

9780 

42 

*7 

8i63 

82i5 

8267 

8320 

8372 

8425 

42 

18 

9831 

9882 

9933 

9984 

..36 

..87 

4i 

18 

8477 

853o 

8582 

8635 

8687 

874o 

4i 

T9 

io.28oi38 

0189 

0240 

0292 

o343 

o394 

4o 

J9 

8792 

8845 

8897 

8949 

9OO2 

9o54 

4o 

20 

io.28o445 

0496 

o548 

0599 

o65o 

0701 

39 

20 

io.299io7 

9l59 

9212 

9264 

93i7 

937o 

39 

21 

0752 

o8o4 

o855 

0906 

0957 

1009 

38 

21 

9422 

9475 

95a7 

958o 

9632 

9685 

38 

22 

1060 

mi 

1162 

I2l4 

1265 

i3i6 

37 

22 

9737 

979° 

9842 

9895 

9947 

37 

23 

i367 

1419 

1470 

l52I 

1572 

1624 

36 

23 

io.3ooo53 

oio5 

oi58 

O2IO 

0263 

o3i5 

36 

24 

i675 

1726 

1777 

1829 

1880 

1931 

35 

24 

o368 

0421 

o473 

o526 

0578 

o63i 

35 

25 

1983 

2034 

2o85 

2137 

2188 

2239 

34 

25 

o684 

o736 

0789 

o84i 

o894 

o947 

34 

26 

2291 

2342 

23g3 

2445 

2496 

2547 

33 

26 

0999 

1052 

no5 

1157 

1210 

1263 

33 

27 

2599 

265o 

2701 

2753 

2804 

2855 

32 

27 

i3i5 

i368 

1421 

i473 

i526 

i579 

32 

28 

2907 

2958 

3009 

3o6i 

3lI2 

3i64 

3i 

28 

i63i 

1  684 

i737 

1789 

1842 

i895 

3i 

29 

32i5 

3266 

33i8 

3369 

3421 

3472 

3o 

29 

1947 

2OOO 

2o53 

2106 

2i58 

2211 

3o 

3o 

ic.  283523 

3575 

3626 

3678 

3729 

3780 

29 

3o 

IO.  3O2264 

23i6 

2369 

2422 

2475 

2527 

29 

3i 

3832 

3883 

3g35 

3986 

4o38 

4089 

28 

3i 

258o 

2633 

2686 

2738 

279I 

2844 

28 

32 

4i4o 

4192 

4243 

4296 

4346 

4398 

27 

32 

2897 

2950 

30O2 

3o55 

3io8 

3i6i 

27 

33 

4449 

45oi 

4552 

46o4 

4655 

4707 

26 

33 

32i3 

3266 

3319 

3372 

3425 

3478 

26 

34 

4?58 

48io 

486i 

49  1  3 

4964 

5oi6 

25 

34 

353o 

3583 

3636 

3689 

3742 

3794 

25 

35 

5067 

5ii9 

5170 

5iJ22 

5273 

5325 

24 

35 

3847 

3900 

3953 

4oo6 

4o59 

4lI2 

24 

36 

5376 

5428 

5479 

553i 

5582 

5634 

23 

36 

4i64 

42  I  7 

427o 

4323 

4376 

4429 

23 

37 

5686 

5737 

5789 

584o 

5892 

5943 

22 

37 

4482 

4535 

4588 

464o 

4693 

4746 

22 

38 

5995 

6o46 

6098 

6i5o 

6201 

6253 

21 

38 

4799 

4852 

49o5 

4958 

5on 

5o64 

21 

39 

63o4 

6356 

64o8 

6459 

65n 

6562 

2O 

39 

6117 

5i7o 

5223 

5276 

5328 

538i 

2O 

4o 

10.286614 

6666 

6717 

6769 

6821 

6872 

19 

4o 

io.3o5434 

5487 

554o 

5593 

5646 

5699 

I9 

4i 

6924 

6975 

7027 

7079 

7i3o 

7182 

18 

4i 

5752 

58o5 

5858 

59n 

5964 

6oi7 

18 

42 

7234 

7285 

7337 

7389 

744o 

7492 

17 

42 

6o7o 

6i23 

6i76 

6229 

6282 

6335 

'7 

43 

7544 

7595 

7647 

7699 

775i 

7802 

16 

43 

6388 

644i 

6494 

6547 

6600 

6654 

16 

44 

7854 

7906 

7957 

8009 

8061 

8n3 

i5 

44 

67o7 

676o 

68i3 

6866 

69i9 

6972 

15 

45 

8i64 

8216 

8268 

8319 

837i 

8423 

i4 

45 

7O25 

7o78 

7i3i 

7i84 

7237 

729o 

i4 

46 

8475 

8526 

8578 

863o 

8682 

8733 

i3 

46 

7344 

7397 

745o 

75o3 

7556 

76o9 

i3 

47 

8785 

8837 

8889 

894i 

8992 

9044 

12 

47 

7662 

77i5 

7768 

7822 

7875 

7928 

12 

48 

9096 

9i48 

9199 

925i 

o^o3 

9355 

II 

48 

7981 

8o34 

8087 

8i4i 

8i94 

8247 

II 

49 

9407 

9458 

gSio 

9562 

9614 

9666 

10 

49 

83oo 

8353 

84o6 

846o 

85i3 

8566 

10 

5o 

10.289718 

9769 

9821 

9873 

9925 

9977 

9 

5o 

10.308619 

8672 

8726 

8779 

8832 

8885 

9 

5i 

10.  290029 

0081 

Ol32 

0184 

0236 

0288 

8 

5i 

8938 

8992 

9o45 

9o98 

9i5i 

9205 

8 

52 

o34o 

o392 

o444 

0496 

o547 

0599 

7 

52 

9258 

93n 

9364 

94i8 

947i 

9524 

7 

53 

o65i 

o7o3 

0755 

0807 

0859 

0911 

6 

53 

9577 

963i 

9684 

9737 

979° 

9844 

6 

54 

0963 

ioi5 

1066 

1118 

1170 

1222 

5 

54 

9897 

995o 

...4 

..57 

.  no 

.164 

5 

55 

1274 

i326 

1  378 

i43o 

1482 

i534 

4 

55 

IO.3lO2I7 

O27O 

o324 

o377 

o43o 

o484 

4 

56 

i586 

i638 

1690 

1742 

1794 

i846 

3 

56 

o537 

o59o 

o644 

o697 

0750 

0804 

3 

57 

1898 

igSo 

2OO2 

2o54 

2106 

2i58 

2 

57 

o857 

0910 

o964 

ioi7 

1070 

1124 

2 

58 

22IO 

2262 

23l4 

2366 

2418 

2470 

I 

58 

u77 

I23l 

1284 

i337 

i39i 

i444 

I 

59 

2522 

2574 

2626 

2678 

2730 

2782 

O 

59 

1498 

i55i 

i6o5 

i658 

1711 

i765 

O 

60" 

50" 

40" 

30" 

20" 

10" 

a 

60" 

50" 

40" 

30" 

20" 

10" 

d 

Co-tangent  of  27  Degrees. 

.3 

Z 

Co-tangent  of  26  Degrees. 

pp   (  1"  2"  3"  4"  5"  (>"  7"  8"  9"  |l      t$l"  2"  3"  4"  5"  6"  7"  8"  9" 
J  5  10  15  21  20  31  33  41  46  !     "J  5  11  16  21  26  32  37  42  47 

L,  O  e  A  It  I  T  H  M  I  C      JSlNES. 


.s 

Sine  of  64  Degrees.      , 

.5 

Sine  of  65  Degrees. 

s 

0" 

10" 

20" 

30" 

40" 

50" 

s 

0" 

10" 

20" 

30" 

40" 

50" 

o 

9.953660 

367o 

368i 

369i 

3701 

3?I2 

59 

o 

9.957276 

7286 

7295 

73o5 

73l5 

7325 

? 

i 

3722 

3732 

3742 

3753 

3763 

3773 

58 

i 

7335 

7344 

7354 

7364 

7384 

CO 

r 

3783 

3794 

38o4 

38i4 

3824 

3835 

57 

2 

7393 

74o3 

74i3 

742S 

7433 

7442 

57 

C 

3845 

3855 

3865 

3876 

3886 

3896 

56 

q 

7452 

7462 

7472 

7482 

7491 

56 

/; 

39o6 

39i7 

3927 

3937 

3947 

3957 

55 

4 

752I 

753i 

754o 

755o 

756o 

55 

5 

3968 

3978 

3988 

3998 

4009 

4oi9 

54 

5 

7570 

7579 

7589 

7599 

76o9 

7619 

54 

6 

4029 

4o39 

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4070 

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53 

6 

7628 

7638 

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7658 

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7677 

53 

7 

4090 

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4i  ii 

4l2I 

4i3i 

4i4i 

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7 

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77i6 

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8 

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8 

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9.957863 

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49 

ii 

4335 

4345 

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48 

ii 

7921 

793i 

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795o 

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48 

12 

4396 

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12 

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8018 

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47 

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4620 
468i 

463o 
469i 

44 
43 

i5 
16 

8i54 
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8i64 
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8i74 

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8i83 
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8i93 

825i 

8203 
8261 

44 
43 

17 

47oi 

47n 

4721  4.732 

4742 

4752 

42 

17 

8271 

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83i9 

42 

18 

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4802 

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18 

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8339 

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8368 

8377 

4i 

19 

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4863 

4873 

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8387 

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20 

9.954883 

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4924 

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39 

20 

9.958445 

8455 

8464 

8474 

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39 

21 

4944 

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4964  4974 

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4995 

38 

21 

85o3 

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38 

22 

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37 

22 

856i 

857i 

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37 

23 

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5io6 

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36 

23 

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8628 

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8648 

8657 

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36 

24 

5i26 

5i36 

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5i66 

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35 

24 

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8686 

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25 

5i86 

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5207  5217 

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25 

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26 

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27 

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27 

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32 

28 

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54o8 

54i8 

3i 

28 

8908 

89i7 

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8937 

8946 

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3i 

29 

5428 

5438 

5448,5458 

5468 

5478 

3o 

29 

8965 

8975 

8985 

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9004 

9013 

3o 

3o 

9.955488 

5498 

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5528 

5538 

29 

3o 

9.959023 

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9042 

9061 

9071 

29 

3i 

5548 

5559 

5569  5579 

5589 

5599 

28 

3i 

9080 

9090 

9100 

9109 

9119 

9128 

28 

32 

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56i9 

5629  5639 

5649 

5659 

27 

32 

9i38 

9i48 

9l57 

9i67 

9176 

9186 

27 

33 

5669 

5679 

5689  5699 

5709 

57i9 

26 

33 

9195 

92o5 

9215 

9224 

9234 

9243 

26 

34 

5729 

5739 

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5769 

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25 

34 

9253 

9262 

9272 

9282 

9291 

g3oi 

25 

35 

5789 

5799 

5809  5819 

5829 

5839 

24 

35 

9310 

9320 

9329 

9339 

9348 

9358 

24 

36 

5849 

5859 

5869  5879 

5889 

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23 

36 

9368 

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9387 

9396 

94o6 

94i5 

23 

37 

59o9 

59i9 

5929  5939 

5949 

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22 

37 

9425 

9434 

9444 

9453 

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9473 

22 

38 

5969 

5979 

5989:5999 

6009 

6019 

21 

38 

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9492 

95oi 

95n 

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953o 

21 

39 

6029 

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3069 

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39 

9539 

9549 

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9568 

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4o 

9.956089 

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6128 

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4o 

9.959096 

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9625 

9634 

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19 

4i 

6i48 

6  1  58 

61686178 

6!88 

6198 

18 

4i 

9654 

9663 

9673 

9682 

9692 

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18 

4a 

6208 

6218 

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6248 

6258 

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42 

97n 

9720 

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9739 

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1.7 

43 

6268 

6278 

6288^6298 

63o8 

63i7 

16 

43 

9768 

9777 

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9796 

98o6 

98i5 

16 

44 

6327 

6337 

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6367 

6377 

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44 

9825 

9834 

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45 

6387 

6397 

6407  6417 

6427 

6437 

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45 

9882 

9891 

9900 

99io 

9919 

9929 

i4 

46 

6447 

6457 

6466  6476 

6486 

6496 

i3 

46 

9938 

9948 

9957 

9967 

9976 

9986 

i3 

47 

65o6 

65i6 

65266536 

6,546 

6556 

12 

47 

9995 

...5 

..24 

..33 

..43 

12 

48 

6566 

6575 

6585 

6595 

66o5 

66i5 

II 

48 

9.96oo5a 

0061 

Of  7I 

0080 

0090 

oo99 

II 

49 

6625 

6635 

6645 

6655 

6665 

6674 

IO 

49 

0109 

0118 

0128 

oi37 

0147 

oi56 

IO 

5o 

9.  956684 

6694 

6704 

67i4 

6724 

6734 

9 

5o 

9.960165 

oi75 

0184 

oi94 

0203 

02l3 

9 

5i 

6744 

6754 

6763 

6773 

6783 

6793 

8 

5i 

O222 

0232 

0241 

0250 

0260 

O269 

8 

52 

68o3 

68i3 

6823 

6833 

6843 

6852 

7 

52 

0279 

0288 

0298 

o3o7 

o3i7 

o326 

7 

53 

6862 

6872 

6882 

6892 

6902 

6912 

6 

53 

o335 

o345 

o354 

o364 

o373 

o382 

6 

54 

692I 

6931 

6941 

695i 

6961 

697i 

5 

54 

0392 

o4oi 

o4n 

0420 

o43o 

o439 

5 

55 

6981 

3990 

7000 

7010 

7020 

7o3o 

4 

55 

o448 

o458 

o467 

o477 

o486 

o4g5 

4 

56 

7o4o 

7o5o 

7o59 

7069 

7079 

•7089 

3 

56 

o5o5 

o5i4 

o524 

o533 

o542 

o552 

3 

57 

7099 

7io9 

7118 

7128 

7i38 

7i48 

2 

57 

o56i 

o57i 

o58o 

o589 

0599 

0608 

2 

58 

7i58 

7i68 

7177 

7187 

7197 

72O7 

I 

58i    0618 

o627 

o636 

o646 

o655 

o665 

I 

59 

72i7 

7227 

7236 

7246 

7256 

7266 

O 

59 

0674 

o683 

0693 

0702 

0711 

072I 

0 

60" 

50" 

40" 

30" 

20" 

10" 

S* 

60" 

50" 

40" 

30"   20"  |  10" 

d 

Co-sine  of  25  Degrees. 

Co-sine  of  24  Degrees. 

i 

P.  Part  j1^'  22'  3S'  4"  55"  66"  77"  8g'  9g" 

--                   -       L 

PPartJ1"  2"  3"  4<  5"  6"  7"  8"  "" 
irt\  1   2   3   4   5   6   7   8   9 

LOGARITHMIC    TANGENTS. 


.5 

Tangent  of  64  Degrees. 

I 

Tangent  of  65  Degrees. 

s 

0" 

10" 

20" 

30" 

40" 

50" 

* 

0" 

10" 

20" 

30" 

40" 

50" 

0 

io.3n8i8 

1872 

1925 

1979 

2032 

2o85 

59 

o 

io.33i327 

i382 

i437 

l492 

i547 

l6O2 

59 

I 

2i3g 

2192 

2246 

2299 

2353 

2406 

58 

I 

1657 

1712 

i767 

l822 

1877 

I932 

58 

2 

2460 

25i3 

2567 

2620 

2674 

2727 

57 

2 

1987 

2042 

2097 

2i53 

2208 

2263 

57 

3     2781 

2834 

2888 

294l 

2995 

3o48 

56 

3 

23i8 

2373 

2428 

2483 

2538 

2593 

56 

A     3io2 

3i55 

3209 

3263 

33i6 

337o 

55 

4 

2648 

2703 

2758 

28i3 

2868 

2924 

55 

5.     3423 

3477 

353o 

3584 

3637 

3691 

54 

5 

2979 

3o34 

3089 

3i44 

3i99 

3254 

54 

6 

374^ 

3798 

3852 

39o5 

3959 

4oi3 

53 

6 

3309 

3364 

3420 

3475 

353o 

3585 

53 

7 

4o66 

4l2O 

4i73 

4227 

4281 

4334 

52 

7 

364o 

3695 

375i 

38o6 

386i 

39i6 

52 

8 

4388 

4442 

4495 

4549 

46o3 

4656 

5i 

8 

397r 

4026 

4082 

4i37 

4i92 

4247 

5i 

9 

4710 

4764 

48i7 

487i 

4925 

4978 

5o 

9 

4302 

4358 

44i3 

4468 

4523 

4579 

5o 

10 

io.3i5o32 

5o86 

SiSg 

5i93 

5247 

53oo 

49 

IO 

io.334634 

4689 

4744 

48oo 

4855 

49io 

49 

ii 

5354 

54o8 

546i 

55!5 

5569 

5623 

48 

ii 

4965 

5021 

5o76 

5i3i 

5i86 

5242 

48 

12 

5676 

5780 

5784 

5838 

589i 

5945 

47 

12 

5s97 

5352 

54o8 

5463 

55i8 

5574 

47 

i3 

5999 

6o53 

6106 

6160 

6214 

6268 

46 

i3 

5629 

5684 

5y4o 

5795 

585o 

59o6 

46 

i4 

632i 

6375 

6429 

6483 

6537 

659o 

45 

i4 

596i 

6016 

6o72 

6l27 

6182 

6238 

45 

i$ 

6644 

6698 

6752 

6806 

6860 

69i3 

44 

i5 

6293 

6349 

64o4 

6459 

65i5 

657o 

44 

16 

6967 

7021 

7075 

7129 

7i83 

7236 

43 

16 

6625 

6681 

6736 

6792 

6847 

69o3 

43 

i? 

7290 

7344 

7398 

7452 

75o6 

756o 

42 

*7 

6958 

7013 

7o69 

7124 

7180 

7235 

42 

18 

7613 

7667 

7721 

7775 

7829 

7883 

4i 

18 

•7291 

7346 

74o2 

7457 

75i3 

7568 

4i 

J9 

7937 

7991 

8o45 

8o99 

8i53 

8206 

4o 

r9 

7624 

7679 

7735 

779° 

7846 

79oi 

4o 

20 

10.318260 

83!4 

8368 

8422 

8476 

853o 

39 

20 

10.337957 

8012 

8068 

8i23 

8179 

8234 

39 

21 

8584 

8638 

8692 

8  746 

8800 

8854 

38 

21 

8290 

8345 

84oi 

8456 

85i2 

8568 

38 

22 

8908 

8962 

9016 

9070 

9I24 

9178 

3? 

22 

8623 

8679 

8734 

8790 

8845 

89oi 

3? 

23 

9282 

9286 

934o 

9394 

9448 

9502 

36 

23 

8957 

9012 

9068 

9123 

9179 

9235 

36 

24 

9556 

9610 

9664 

97i8 

9772 

9826 

35 

24 

9200 

9346 

9402 

9457 

95i3 

9569 

35 

25 

9880 

9934 

9988 

..42 

..96 

.161 

34 

25 

9624 

9680 

9735 

9791 

9847 

99O2 

34 

26 

IO.32O2O5 

0259 

o3i3 

o367 

0421 

o475 

33 

26 

9958 

..14 

..7o 

.125 

.181 

.237 

33 

27 

0529 

o583 

0637 

0692 

0746 

0800 

32 

27 

I0.340292 

o348 

o4o4 

o46o 

o5i5 

o57i 

32 

28 

o854 

0908 

0962 

1016 

1071 

1125 

3i 

28 

0627 

0682 

o738 

o794 

o85o 

o9o5 

3i 

29 

1179 

1233 

1287 

i34i 

i396 

i45o 

3o 

29 

0961 

1017 

io73 

1129 

n84 

1240 

3o 

3o 

*o  32i5o4 

i558 

1612 

1666 

1721 

1775 

29 

3o 

IO.34I2Q6 

i352 

i4o8 

i463 

iSig 

i575 

29 

3i 

1829 

i883 

I938 

1992 

2046 

2100 

28 

3i 

i63i 

1687 

I742 

1798 

i854 

I9IO 

28 

32 

2i54 

2209 

2263 

23i7 

2371 

2426 

27 

32 

1966 

2022 

2078 

2I33 

2189 

2245 

27 

33 

2480 

2534 

2588 

2643 

2697 

275l 

26 

33 

2301 

2357 

24l3 

2469 

2525 

258i 

26 

34 

2806 

2860 

2914 

2968 

3o23 

3o77 

25 

34 

2636 

2692 

2748 

2804 

2860 

29i6 

25 

35 

3i3i 

3i86 

324o 

3294 

3349 

34o3 

24 

35 

2972 

3028 

3o84 

3i4o 

3i96 

3252 

24 

36 

3457 

35i2 

3566 

3620 

3675 

3729 

23 

36 

33o8 

3364 

3420 

3476 

3532 

3588 

23 

37 

3783 

3838 

3892 

3947 

4ooi 

4o55 

22 

37 

3644 

3700 

3756 

38i2 

3868 

3924 

23 

38 

4no 

4i64 

4219 

4273 

4327 

4382 

21 

38 

3980 

4o36 

4092 

4i48 

4204 

4260 

21 

39 

4436 

449i 

4545 

4599 

4654 

47o8 

20 

39 

43i6 

4372 

4428 

4484 

454o 

4596 

2O 

4o 

10.324763 

48i7 

4872 

4926 

498i 

5o35 

19 

4o 

io.344652 

47o8 

4764 

4821 

4877 

4933 

J9 

4i 

5089 

5i44 

5198 

5253 

5307 

5362 

18 

4i 

4989 

5o45 

5ioi 

5i57 

52i3 

5269 

18 

42 

54i6 

547i 

5525 

558o 

5634 

5689 

i? 

42 

5326 

5382 

5438 

5494 

555o 

56o6 

II 

43 

6743 

5798 

5852 

5907 

5962 

6016 

16 

43 

5663 

57i9 

5775 

583i 

5887 

5943 

16 

44 

6071 

6i25 

6180 

6234 

6289 

6343 

i5 

44 

6000 

6o56 

6112 

6168 

6224 

628! 

i5 

45 

6398 

6453 

65o7 

6562 

6616 

6671 

i4 

45 

6337 

6393 

6449 

65o6 

6562 

6618 

i4 

46 

6726 

6780 

6835 

6889 

6944 

6999 

i3 

46 

6674 

6731 

6787 

6843 

6899 

6956 

i3 

47 

7o53 

7108 

7162 

7217 

7272 

7326 

12 

47 

7012 

7068 

7I25 

7181 

7237 

7293 

12 

48 

738i 

7436 

7490 

7545 

7600 

7654 

II 

48 

735o 

74o6 

7462 

75l9 

7575 

763i 

II 

49 

7709 

7764 

7818 

7873 

7928 

7982 

IO 

49 

7688 

7744 

78oo 

7857 

79i3 

7969 

IO 

5o 

10.328037 

8092 

8i47 

8201 

8256 

83n 

g 

5o 

10.348026 

8082 

8i39 

8i95 

825i 

83o8 

9 

5i 

8365 

8420 

8475 

853o 

8584 

8639 

8 

5i 

8364 

8421 

8477 

8533 

859o 

8646 

8 

52 

8694 

8749 

88o3 

8858 

89i3 

8968 

7 

52 

87o3 

8759 

88i5 

8872 

8928 

8985 

7 

53 

9023 

9077 

9182 

9187 

9242 

9297 

6 

53 

9041 

9o98 

9i54 

9211 

9267 

9324 

6 

54 

935i 

9406 

946i 

95i6 

9571 

9625 

5 

54 

9380 

9437 

9493 

955o 

96o6 

9663 

5 

55 

9680 

9735 

979° 

9845 

9900 

9955 

4 

55 

9719 

9776 

9832 

9889 

9945 

.  .  .2 

4 

56 

10.330009 

006^ 

0119 

0174 

0229 

0284 

3 

56 

io.35oo58 

on5 

OI7I 

0228 

0285 

o34i 

3 

57 

oSSg 

089^ 

o448 

o5o3 

o558 

o6i3 

2 

57 

o398 

o454 

o5n 

o567 

0624 

0681 

2 

58 

0668 

0723 

0778 

o833 

0888 

o943 

I 

58 

o737 

o794 

o85o 

o9o7 

o964 

IO2O 

I 

59 

0998 

io53 

1108 

n63 

1218 

1272 

O 

59 

1077 

n34 

II9O 

I247 

i3o4 

i36o 

O 

60"    |  50" 

40" 

30" 

20"   10" 

j 

60" 

50" 

40"  |  30" 

20" 

10" 

a' 

Co-tangent  of  25  Degrees. 

a 

Co-tangent  of  24  Degrees. 

§ 

p  p  t$  1"  2"  3"  4"  5"  6"  7"  8'1  9" 

C  1"  2"  3"  4"  5"  6"  7"  8"  9" 

•j  5  11  LS  22  27  33  38  43  49 

in\  6  Jl  17  22  28  33  39  45  50 

LOGARITHM  i  ;    SINES. 


d 

Sine  of  66  Degrees. 

d 

Sine  of  67  Degrees. 

is 

0"     10" 

20" 

30" 

40" 

50" 

8 

0" 

10"   20" 

30" 

40" 

50" 

o 

9  .960780 

o74o 

Q749 

0758 

0768 

°777 

59 

0 

9  .964026 

4o35 

4o44 

4o53 

4o62 

4071 

59 

i 

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55 

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4 

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58 
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I 
0 

GO" 

50" 

40" 

30"   20" 

10" 

ci 

60" 

50" 

40" 

30"   20" 

10" 

a 

Co-sine  of  23  Degrees. 

s 

Co-sine  of  22  Degrees. 

2 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 
t'.Fart^  J23455678 

P.  Part  J  l"  ~"  3"  4"  5"  6"  7"  8"  y" 

LOGARITHMIC    TAXGENTS. 


91 


1  — 
A 

Tangent  of  66  Degrees. 

c 

Tangent  of  67  Degrees. 

9 

0" 

10" 

20" 

30" 

40" 

50" 

2 

0" 

10" 

2<y 

30" 

40*  )  50" 

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io.35i4i7 

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0 

10.372148 

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59 

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1757 

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1870 

1927 

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2040 

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2499 

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2617 

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58 

2 

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57 

2 

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2968 

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57 

3 

2438 

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2608 

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56 

3 

3203 

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3320 

3379 

3437 

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56 

4 

2778 

2835 

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55 

4 

3555 

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3672 

373, 

3789 

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55 

5 

3119 

3:76 

3233 

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54 

5 

39°7 

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54 

6 

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53 

6 

4259 

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53 

7 

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7 

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8 

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9 

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10 

10  354826 

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20 

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20 

10.379213 

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21 

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28 

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29 

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29 

2418 

2478 

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10.361698 

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22 

37 

5282 

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22 

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21 

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564i 

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0 

59 

3227 

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O 

60" 

50" 

40" 

30"  |  20" 

10" 

£3 

60" 

50"  1  40" 

30" 

20" 

10" 

A 

Co-tangent  of  23  Degrees. 

s 

Co-tangent  of  22  Degrees.    si 

P  Parf5  l"  2"  3"   4"  5//   6//  7"   8"  9//  M        tf1"  2"   3//   4//   5"  6"  7"  8"  9V 

iri$  6  12  17  23  29  35  40  46  52      irt}  6  12  18  24  30  36  42  48  54 

_  *.  1 

LOGARITHMIC     !S  I  \  E  S. 


jj 

Sine  of  68  Degrees. 

.5 

Sine  of  69  Degrees. 

* 

0" 

10" 

20"   30"  |  40" 

50" 

S 

0"    |  10" 

20" 

30" 

40" 

50" 

o 

9.9C-7i66 

7174 

7l83  7I9I 

72OO 

7208 

59 

o 

9  970152 

0160 

Ol68 

0176 

0184 

0192 

59 

I 

7217 

7225 

7234  7242 

725l 

7259 

58 

i 

02OO 

0208 

02l6 

022^ 

0233 

024l 

58 

2 

7268 

7276 

7285  7293 

7302 

73io 

57 

2 

0249 

0257 

0265 

0273 

0281 

0289 

57 

3 

73i9 

7327 

7336  7344 

7353 

736i 

56 

3 

0297 

o3o5 

o3i3 

0321 

0329 

o337 

56 

4 

737o 

7378 

7387  7395 

74o4 

7412 

55 

4 

o345 

o353 

o36i 

0370 

o378 

o386 

55 

e 

742I 

-7429 

7437  7446 

7454 

7463 

54 

c 

o394 

O4O2 

o4io 

0418 

0426 

o434 

54 

6 

747i 

748o 

7488  7497 

75o5 

75i4 

53 

6 

0442 

o45o 

o458 

o466 

0474 

0482 

53 

7 

7522 

753i 

7539  7547 

7556 

7564 

52 

7 

o49o 

0498 

o5o6 

o5i4 

O522 

o53o 

52 

8 

7573 

758i 

759o  7598 

76o7 

76i5 

5i 

8 

o538 

o546 

o554 

o562 

0570 

o578 

5i 

9 

7624 

7632 

764o  7649 

7657 

7666 

5o 

9 

o586 

o594 

o6o3 

0611 

0619 

0627 

5o 

10 

9-967674 

7683 

7691 

7699 

77o8 

7716 

49 

IO 

9.97o635 

o643 

o65i 

o659 

0667 

o675 

49 

ii 

77a5 

7733 

7742  7750 

7758 

7767 

48 

ii 

o683 

0691 

0699 

0707 

07l5 

072^ 

48 

12 

7775 

7784 

7792  7801 

7809 

7817 

47 

12 

0731 

o739 

0747 

o755 

0763 

0771 

47 

i3 

-7826 

7843  785i 

7859 

7868 

46 

i3 

0779 

o787 

0795 

o8o3 

0811 

0819 

46 

i4 

7876 

7885 

7893 

79oi 

7910 

7918 

45 

i4 

0827 

o835 

0842 

o85o 

o858 

0866 

45 

i5 

7927 

7935 

7943 

7952 

7960 

7969 

44 

i5 

08  74 

0882 

0890 

o898 

0906 

09M 

44 

16 

7977 

7985  7994 

8002 

8011 

8019 

43 

16 

O922 

0930 

o938 

o946 

0954 

0962 

43 

17 

802-7 

8o368o448o53 

8061 

8069 

42 

17 

0970 

o978 

0986 

o994 

1002 

IOIO 

42 

18 

8o78 

808680948103 

8m 

8120 

4i 

18 

1018 

1026 

io34 

IO42 

io5o 

io58 

4i 

19 

8128 

8i368i458i53 

8161 

8170 

4o 

i9 

1066 

io73 

1081 

io89 

1097 

no5 

4o 

20 

9.968i78 

81878195  8203 

8212 

8220 

39 

20 

9.97in3 

II2I 

1129 

n37 

n45 

ii53 

39 

21 

8228 

8237 

8245 

8253 

8262 

8270 

38 

21 

1161 

1169 

1177 

n85 

1193 

I2OO 

38 

22 

8278 

8287' 

8295 

83o3 

83i2 

8320 

37 

22 

1208 

1216 

1224 

1232 

1240 

1248 

37 

23 

8329 

8337!8345 

8354 

8362 

83-n 

36 

23 

1256 

1264 

1272 

1280 

1288 

I296 

36 

24 

8379 

8387'8395 

84o4 

8412 

8420 

35 

24 

i3o3 

i3n 

1319 

l327 

i335 

1  343 

35 

25 

8437 

8445  8454 

8462 

8470 

34 

25 

i35i 

i359 

i367 

i375 

i383 

1390 

34 

26 

8479 

8487 

8495l85o3 

85i2 

8520 

33 

26 

i398 

i4o6 

i4i4 

1422 

i43o 

i438 

33 

27 

8528 

8537 

85458553 

8562 

857o 

32 

27 

1  446 

i454 

1462 

i469 

1477 

i485 

32 

28 

8578 

8587 

8595 

86o3 

8612 

8620 

3i 

28 

1493 

i5oi 

iSog 

I5i7 

i525 

i532 

3i 

29 

8628 

8636 

8645 

8653 

866! 

867o 

3o 

29 

i54o 

1  548 

i556 

1  564 

1572 

i58o 

3o 

3o 

9.968678 

86868694 

87o3 

8711 

8-719 

29 

3o 

9.971588 

i595 

i6o3 

1611 

1619 

1627 

29 

3i 

8728 

87368744 

8752 

8761 

8769 

28 

3i 

i635 

1  643 

i65i 

i658 

1666 

1674 

28 

32 

8777 

8786 

8794 

8802 

8810 

8819 

27 

32 

1682 

i69o 

1698 

I7o6 

1713 

1721 

27 

33 

8827 

8835  8844 

8852 

8860 

8868 

26 

33 

1-729 

1737 

1745 

i753 

1761 

1768 

26 

34 

8877 

88858893 

8901 

8910 

8918 

25 

34 

1776 

1784 

1792 

1800 

1808 

i8i5 

25 

35 

8926 

89348943 

895i 

8959 

8967 

24 

35 

1823 

i83i 

i839 

i847 

i855 

1862 

24 

36 

8976 

89848992 

9000 

9oo9 

9017 

23 

36 

1870 

1878 

1886 

1894 

1902 

1909 

23 

37 

9025 

9o33  9042 

goSo 

9o58 

9066 

22 

37 

1917 

!925 

i933 

1941 

1949 

1956 

22 

38 

9o75 

9083  9091 

9099 

9108 

9116 

21 

38 

1964 

I972 

i98o 

1988 

i995 

2003 

21 

39 

9124 

9132  9141 

9149 

9i57 

9i65 

2O 

39 

2011 

20I9 

2027 

2034 

2042 

2o5o 

2O 

4o 

9.969173 

9182 

9190 

9198 

92O6 

9215 

19 

4o 

9.972o58 

2066 

2073 

2081 

2089 

2097 

I9 

4i 

9223 

923i: 

9239 

9247 

9256 

9264 

18 

4i 

2IO5 

2112 

2120 

2128 

2i36 

2i44 

18 

42 

9272 

92809288 

9297 

93o5 

93i3 

17 

42 

2l5l 

2i59 

2167 

<i75 

2i83 

2190 

17 

43 

9321 

9329*9338 

9346 

9354 

9362 

16 

43 

2198 

22O6 

22l4 

2221 

2229 

2237 

1  6 

44 

937o 

9379 

9387 

9395 

94o3 

9411 

i5 

44 

2245 

2253 

226o 

2268 

2276 

2284 

i5 

45 

9420 

9428 

9436 

9444 

9452 

946i 

i4 

45 

2291 

2299 

2307 

23i5 

2322 

233o 

i4 

46 

9469 

9477 

9485 

9493 

95oi 

95io 

i3 

46 

23382346 

2354 

236i 

2369 

2377 

i3 

47 

95i8 

95269534 

9542 

955o 

9559 

12 

47 

2385 

2392 

2400 

2408 

2416 

2423 

12 

48 

9567 

9575 

9583 

959i 

9599  9608 

II 

48 

243  1 

2439 

2447 

2454 

2462 

2470 

II 

9616 
9.969665 

96739681 

964o 
9689 

9648  9657 

IO 

9 

I9 
5o 

2478 
9.972524 

2485 

2532 

2493 
2539 

25oi 

2547 

25o8 
2555 

25i6 
2563 

IO 

5i 

9714 

97229780 

9738 

9746  9754 

8 

5i 

2570 

2578 

2586 

2593 

2601 

2609 

8 

52 

9762 

977i  9779 

9787 

9795  98°3 

7 

52 

2617 

2624 

2632 

2640 

2648 

2655 

7 

53 

9811 

98i99828 

9836 

98449852 

6 

53 

2663 

2671 

2678 

2686 

2694 

2701 

6 

54 

9860 

98689876 

9884 

98939901 

5 

54 

2709 

2717 

2725 

2732 

2740 

2748 

5 

55 

9909 

99i79925 

9933 

9941  9949 

4 

55 

2755 

2763 

2771 

2778 

2786 

2794 

4 

56 

9957 

9966  9974 

9982 

9990  9998 

3 

56 

2802 

2809 

2817 

2825 

2832 

2840 

3 

57 

).  970006 

ooi4 

OO22 

oo3o 

oo38  oo47 

2 

57 

2848 

2855 

2863 

2871 

2878 

2886 

2 

58 
59 

oo55 
oio3 

oo63 

OIII 

007I 
Olig 

oo79 

OI27 

008-7  OO95 
oi36oi44 

0 

58 
59 

2894 
2940 

2901 
2947 

290929I7 

2g55  2963 

2924  2932 
29702978 

ol 

60" 

50" 

40"   30"  I  20"   10" 

ft 

60"     50"   40"   30"   20"   10"  |  rf 

Co-sine  of  21  Degrees. 

.a 

Co-sine  of  20  Degrees.    |  * 

A  1"  2"  3"  4"  5"  6"  7"  S"  9"  ||  „  ,,  ,f  1"  2"  3//  4"  5"  6"  7"  8"  S" 
vt)  1   2   2   3   4   5   6   7   7  II  P'Part}  1   2234   i  R   667 

LOGARITHMIC    TANGENTS. 


1 

Tangent  of  68  Degrees. 

.3 

Tangent  of  69  Degrees. 

§ 

0" 

10"  |  20"  |  30"  |  40"  ]  50"  | 

2 

0" 

10"  |  20" 

30" 

40" 

50" 

0 

I 

io.39359o 
3954 

365i  37i2 
4oi5  4076 

3772i3833 
4i36l4i97 

3894 
4258 

59 

58 

0 

I 

io.4i5823 
6200 

5886 
6263 

5948 
6326 

6011 
6389 

6074 

6452 

6i37 
65i5 

r 

CO 

2 

43i8 

4379444o45oo 

456i 

4622 

5y 

2 

6578 

664i 

67o4 

6767 

683o 

6893 

57 

3 

4683 

4743 

48o4 

4865 

4926 

4986 

56 

3 

6956 

7O2O 

7o83 

7i46 

7200 

7272 

56 

4 

5o47 

5io8 

5169 

5229 

529O 

535i 

55 

4 

7335 

7398 

746i 

7524 

7587 

765o 

55 

5 

5412 

5473 

5533 

5594 

5655 

57i6 

54 

5 

77i4 

7777 

784o 

79°3 

7966 

8029 

54 

6 

5777 

5838 

5898 

5959 

6020 

6081 

53 

6 

8o93 

8i56 

82I9 

8282 

8345 

84o9 

53 

7 

6142 

6203 

6264 

6325 

6385 

6446 

52 

7 

8472 

8535 

8598 

8661 

8725 

8788 

52 

8" 

0  JO7 

6568 

6629 

6690 

675i 

6812 

5i 

8 

885i 

89i4 

8978 

9o4i 

9io4 

9i68 

5i 

9 

6873 

6934 

6995 

7o56 

7117 

7178 

5o 

9 

923l 

9294 

9358 

942I 

9484 

9547 

5o 

10 

io.397239 

73oo 

736i 

7422 

7483 

7544 

49 

10 

10.419611 

9674 

9738 

98oi 

9864 

9928 

49 

ii 

76o5 

7666 

7727 

7788 

7849 

79io 

48 

ii 

9991 

..54 

.118 

.181 

.245 

.3o8 

48 

12 

7971 

8o32 

8o93 

8i54 

82i5 

8276 

47 

12 

I0.42037I 

o435 

0498 

o562 

0625 

o689 

47 

i3 

8337 

8399 

846o 

852i 

8582 

8643 

46 

i3 

O752 

0816 

0879 

o943 

1006 

1070 

46 

i4 

8704 

8765 

8826 

8888 

8949 

9oio 

45 

i4 

n33 

ii97 

1260 

1324 

i387 

i45i 

45 

i5 

0.071 

9l32 

9194 

9255 

93i6 

9377 

44 

i5 

i5i4 

i578 

i64i 

1705 

i769 

i832 

44 

16 

9438 

95oo 

956i 

9622 

9683 

9744 

43 

16 

1896 

i959 

2023 

2086 

2i5o 

22l4 

43 

17 

98o6 

9867 

9928 

9989 

.  .5i 

.  112 

4s 

17 

2277 

234i 

24o5 

2468 

2532 

2596 

42 

18 

10.400173 

0235 

0296 

o357 

o4i9 

o48o 

4i 

18 

2659 

2723 

2787 

285o 

2914 

2978 

4i 

i9 

o54i 

0602 

o664 

0725 

0787 

o848 

4o 

19 

3o4i 

3io5 

3l69 

3233 

3296 

336o 

4o 

20 

io.4oo9o9 

0971 

1032 

io93 

n55 

1216 

39 

20 

10.42.3424 

3488 

355j 

36i5 

3679 

3743 

39 

21 

1278 

1339 

i4oo 

1462 

i523 

i585 

38 

21 

3807 

387o 

3934 

3998 

4062 

4126 

38 

22 

1  646 

1707 

1-769 

i83o 

l892 

i953 

37 

22 

4190 

4253 

43i7 

438i 

4445 

45o9 

37 

23 

20l5 

2076 

2i38 

2I99 

2261 

2322 

36 

23 

4573 

4637 

4701 

4764 

4828 

4892 

36 

24 

2384 

2445 

25o7 

2568 

263o 

269I 

35 

24 

4g56 

5O2O 

5o84 

5i48 

5212 

5276 

35 

25 

2753 

28i5 

2876 

2938 

2999 

3o6i 

34 

25 

534o 

54o4 

5468 

5532 

5596 

566o 

34 

26 

3122 

3i84 

3246 

3307 

3369 

343o 

33 

26 

5724 

5788 

5852 

59i6 

598o 

6o44 

33 

27 

3492 

3554 

36i5 

3677 

3739 

38oo 

32 

27 

6108 

6172 

6236 

63oo 

6364 

6429 

32 

28 

3862 

3924 

3g85 

4o47 

4io9 

4170 

3i 

28 

6493 

6557 

6621 

6685 

6749 

68i3 

3i 

29 

4232 

4294 

4356 

44i7 

4479 

454i 

3o 

29 

6877 

694i 

7006 

7070 

7i34 

7198 

3o 

3o 

io.4o46o2 

4664 

^726 

4788 

485o 

49n 

29 

3o 

10.427262 

7327 

739i 

7455 

75i9 

7583 

29 

3i 

4973 

5o35 

5097 

5i58 

522O 

5282 

28 

3i 

7648 

7712 

7776 

784o 

79o5  7969 

28 

32 

5344 

54o6 

5468 

5529 

559i 

5653 

27 

32 

8o33 

8097 

8162 

8226 

8290 

8355 

27 

33 

57i5 

5777 

5839 

59oi 

5962 

6024 

26 

33 

8419 

8483 

854.8 

8612 

8676 

874i 

26 

34 

6086 

6i48 

6210 

6272 

6334 

6396 

25 

34 

88o5 

8869 

8934 

8998 

9062 

9I27 

25 

35 

6458 

652O 

6582 

6644 

67o6 

6768 

24 

35 

9191 

9256 

9320 

9384 

9449 

95i3 

24 

36 

6829 

6891 

6953 

7oi5 

7077 

7i39 

23 

36 

9578 

9642 

97°7 

977i 

9836 

99oo 

23 

37 

7201 

7263 

7326 

7388 

745o 

75l2 

22 

37 

9965 

..29 

..94 

.i58 

.223 

.287 

22 

38 

7574 

7636 

7608 

7760 

7822 

7884 

21 

38 

io.43o352 

o4i6 

o48i 

o545 

0610 

o674 

21 

39 

7946 

8008 

8070 

8i32 

8i95 

8257 

2O 

39 

o739 

o8o3 

0868 

o933 

°997 

1062 

2O 

4o 

io.4o83i9 

838i 

8443 

85o5 

8567 

863o 

I9 

4o 

I0.43lI27 

n9i 

1256 

1320 

i385 

i45o 

19 

4i 

8692 

8754 

8816 

8878 

894o 

9oo3 

18 

4i 

i5i4 

l579 

1  644 

1708 

i773 

i838 

18 

42 

9o65 

9127 

9189 

9252 

93i4 

9376 

iy 

42 

I9O2 

1967 

2032 

2097 

2161 

2226 

17 

43 

9438 

gSoi 

9563 

9625 

9687 

975o 

16 

43 

229I 

2356 

2420 

2485 

255o 

26i5 

16 

44 

98l2 

9874 

9937 

9999 

..61 

.123 

i5 

44 

2680 

2744 

2809 

2874 

2939 

3oo4 

i5 

45 

10.410186 

0248 

o3io 

o373 

o435 

o498 

i4 

45 

3o68 

3i33 

3i98 

3263 

3328 

3393 

i4 

46 

o56o 

0622 

o685 

0747 

o8o9 

o872 

i3 

46 

3458 

3522 

3587 

3652 

37i7 

3782 

i3 

4? 

o934 

0997 

1059 

1122 

1184 

1246 

12 

47 

3847 

39I2 

3977 

4042 

4107 

4l72 

12 

48 

i3o9 

i434 

1496 

i559 

1621 

II 

48 

4237 

4302 

4367 

4432 

4497 

4562 

II 

49 

1  684 

1746 

1809 

1871 

i934 

i996 

10 

49 

4627 

4692 

4757 

4822 

4887 

4952 

1° 

5o 

io.4i2o59 

2121 

2184 

2246 

23o9 

237I 

9 

5o 

io.435oi7 

5o82 

5i47 

5212 

5277 

5342 

9 

5i 

2434 

2497 

2559 

2622 

2684 

2747 

8 

5i 

54o7 

5473 

5538 

56o3 

5668 

5733 

8 

52 

2810 

2872 

2935 

2997 

3o6o 

3i23 

7 

52 

5798 

5863 

5929 

5994 

6o59 

6124 

7 

53 

3i85 

3248 

33  1  1 

3373 

3436 

3499 

6 

53 

6i89 

6254 

6320 

6385 

64  5  o 

65i5 

6 

54 

356i 

3624 

3687 

3749 

38i2 

3875 

5 

54 

658i 

6646 

67ii 

6776 

6842 

69o7 

5 

55 

3938 

4ooo 

4o63 

4126 

4i89 

425l 

4 

55 

6972 

7o37 

7io3 

7168 

7233 

7299 

4 

56 

43i4 

4377 

444o 

45o2 

4565 

4628 

3 

56 

7364 

7429 

7495 

756o 

7625 

7691 

3 

57 

469i 

4754 

48i7 

4879 

4942 

5oo5 

2 

57 

7756 

7822 

7887 

7952 

8018 

8o83 

9 

58 

5o68 

5i3i 

5194 

5256 

53i9 

5382 

I 

58 

8i49 

82l4 

8279 

8345 

84io 

8476 

I 

59 

5445 

55o8 

557i 

5634 

5697 

576o 

0 

59 

854i 

86o7 

8672 

8738 

88o3 

8869 

O 

60" 

50" 

40" 

30" 

20" 

10" 

j 

60" 

50" 

40" 

30" 

20"  |  10" 

. 

Co-tangent  of  21  Degrees. 

2 

Co-tangent  of  20  Degrees. 

1 

p  P  t$  1"  2"  3"  4"  5"  6"  7"  8"  9" 

t  1"  2"  3"  4"  5"  6"  7"  8"  9" 

I  6  12  19  25  31  37  43  49  56 

irt)  6  13  19  26  32  39  45  51  58 

Ufo 


liOGABITHMIC     SlNEB. 


a 

Sine  of  72  Degrees. 

| 

Sine  of  73  Degrees. 

55 

0" 

10" 

20"  |  30"   40" 

50" 

0" 

10" 

20" 

30"^ 

40" 

50" 

o 

9.9782o6 

8213 

8220822-7  8234 

8241 

59 

O 

9.98o596 

o6o3 

0609 

0616 

0622 

0628 

59 

I 

8247 

8254 

8261 

82688275 

8282 

58 

I 

o635 

o64i 

o648 

o654 

0661 

066-7 

58 

2 

8288 

8295 

8302 

83o9!83i6 

8322 

57 

2 

0673 

0680 

0686 

o693 

0699 

0-706 

57 

3 

8329 

8336 

8343835o8357 

8363 

56 

3 

0712 

0718 

0725 

0731 

o738 

o744 

56 

4 

837o 

8377 

83848391 

8397 

84o4 

55 

4 

0750 

o757 

o763 

0770 

0776 

o783 

!5 

5 

84  1  1 

84i8 

8425  843i 

8438 

8445 

54 

5 

0789 

o795 

0802 

0808 

o8i5 

0821 

54 

6 

7 

8452 
8493 

8459  8465  8472 
849985o685i3 

8479 
8520 

8486 
852? 

53 

52 

6 

7 

0827 
0866 

o834 
0872 

o84o 
o878 

0847 
o885 

o853 
0891 

o859 
o898 

53 

52 

8 

8533 

854o 

8547  8554856i 

8567 

5i 

8 

0904 

0910 

°9J7 

0923 

ogSo 

o936 

5i 

9 

8574 

858.1 

85888594 

8601 

8608 

5o 

9 

0942 

0949 

o955 

0961 

0968 

o974 

5o 

IO 

9.978615 

8622 

86288635 

8642 

8649 

49 

10 

9.980981 

0987 

0993 

1000 

1006 

1012 

49 

ii 

8655 

8662  8669|8676  8682 

8689 

48 

ii 

1019 

IO25 

io3i 

io38 

io44 

io5i 

48 

12 

8696 

87o387o9;87i68723 

873o 

47 

12 

1057 

io63 

10-70 

1076 

1082 

io89 

47 

1.3 

'  8737 

8743875o!87578764 

877o 

46 

i3 

1095 

IIOI 

1108 

ui4 

II2O 

II27 

46 

i4 

8777 

8784!879i 

8797 

88o4 

8811 

45 

i4 

n33 

1139 

n46 

Il52 

1  1  58 

n65 

45 

i5 

8817 

8824i883i 

8838 

8844 

885i 

44 

i5 

1171 

1177 

n84 

1190 

n96 

1203 

44 

16 

8858 

8865  887i 

8878 

8885 

8892 

43 

16 

1209 

I2l5 

1222 

1228 

1234 

1241 

43 

i7 

8898 

89o5  89i2 

8918 

8925 

8932 

42 

17 

1247 

1253 

1260 

1266 

I272 

I279 

42 

18 

8939 

8945  8952 

8959 

8965 

89-72 

4i 

18 

1285 

1291 

1298 

i3o4 

i3io 

i3i7 

4i 

19 

8979 

8986  8992 

8999 

9oo6 

9012 

4o 

i9 

i323 

1329 

i336 

1  342 

1  348 

1  354 

4o 

20 

9.  979019 

90269033 

9039 

9o46 

9o53 

39 

20 

9.981361 

1367 

i373 

i38o 

1  386 

l392 

39 

21 

9059 

9066  9o73 

9°79 

9o86 

9o93 

38 

21 

i399 

i4o5 

i4n 

1417 

1424  i43o 

38 

22 

9100 

9106  on3 

9I2O 

9I26 

9i33 

37 

22 

i436 

i443 

i449 

i455 

i46i  i468 

37 

23 

9140 

91469153 

9i6o 

9166 

9i73 

36 

23 

i474 

i48o 

i487 

i4g3 

1499 

i5o5 

36 

24 

9180 

9186  9193 

92OO 

9206 

9213 

35 

24 

l5l2 

i5i8 

i524 

i53i 

i537 

i543 

35 

25 

9220 

9227  9233 

9240 

9247 

9253 

34 

25 

i549 

i556 

i562 

i568 

i574 

i58i 

34 

26 

27 

9260 

9267^273 
93069313 

928o 

932O 

9287  9293 
98269333 

33 

32 

26 
27 

1587 
1625 

i593 
:63i 

i599 
i637 

1606 
i643 

1612 
i65o 

1618 
i656 

33 

32 

28 

934o 

93469353 

9360 

93669373 

3i 

28 

1662 

1668 

i675 

1681 

i687 

i693 

3i 

29 

9380 

93869393 

9400 

94o6  94i3 

3o 

29 

1700 

1706 

I7I2 

1718 

1724 

i73i 

3o 

3o 

9.979420 

9426  9433 

9439 

9446 

9453 

29 

3o 

9  98i737 

1743 

i749 

i756 

1-762 

i768 

29 

3i 

9459 

94669473 

9479 

9486 

9492 

28 

3i 

1774 

1781 

1787 

1793 

i799 

1806 

28 

32 

9499 

g5i2 

95l9 

9526 

9532 

27 

32 

18:2 

1818 

1824 

i83o 

i837 

i843 

27 

33 

9539 

9545 

9552 

9559 

9565 

9572 

26 

33 

1849 

i855 

1861 

1868 

i874 

1880 

26 

34 

9579 

9585 

9592 

9598 

9605 

96l2 

25 

34 

1886 

1893 

i899 

1905 

i9n 

1917 

25 

35 

9618 

96259631 

9638 

9645 

965i 

24 

35 

1924 

1930 

i936 

1942 

i948 

i955 

24 

36 

9658 

9664  9671 

9678 

9684 

969i 

23 

36 

1961 

1967 

i973 

19.79 

i986 

I992 

23 

37 

9697 

9704 

9711 

97i7 

9724 

973o 

22 

37 

1998 

2004 

2010 

2016 

2023 

2029 

22 

38 

9737 

9743 

975o 

9757 

9763 

977° 

21 

38 

2o35 

2o4l 

2047 

2o54 

2060 

2O66 

21 

39 

9776 

9783 

979° 

9796 

98o3 

9809 

2O 

39 

2072 

2078 

2084 

2091 

2097 

2103 

2O 

4o 

9.9798i6 

9822 

9829 

9836 

9842 

9849 

'9 

4o 

9.982109 

2Il5 

2122 

2128 

2i34 

2l4o 

19 

4i 

9855 

9862:9868 

9875 

988i 

9888 

18 

4i 

2l46 

2l52 

2l59 

2i65 

2I7I 

2I77 

18 

42 

9895 

99oi 

9908 

9914 

992i 

9927 

17 

42 

2i83 

2189 

2I95 

2202 

2208 

22l4 

17 

43 

9934 

994o  9947 

9954 

996o 

9967 

16 

43 

222O 

2226 

2232 

2239 

2245 

225l 

16 

44 

9973 

998o  9986 

9993 

9999 

...6 

i5 

44 

2257 

2263 

2269 

2275 

2282 

2288 

i5 

45 

9.98ooi2 

ooi9  0026 

OO32 

oo39 

oo45 

i4 

45 

2294 

2300 

23o6 

2312 

23i8 

2324 

i4 

46 

OO52 

oo58  oo65 

007I 

oo78 

oo84 

i3 

46 

233i 

2337 

2343 

2349 

2355 

236i 

i3 

47 

0091 

009-7  oio4 

OIIO 

OII7  OI23 

12 

47 

2367 

2373 

238o 

2386 

2392 

2398 

12 

48 

oi3o 

oi36oi43 

oi49 

oi56  oi63 

II 

48 

2404 

2410 

2416 

2422 

2429 

2435 

II 

49 

0169 

oi76!oi82 

oi89 

OI95,0202 

IO 

49 

2441 

2447 

2453 

2459 

2465 

247I 

IO 

5o 

9.980208 

O2l5 

0221 

0228 

O234  0241 

9 

5o 

9.982477 

2484 

249O 

2496 

2502 

25o8 

9 

Si 

0247 

0254 

O26o 

026-7 

0273  0280 

8 

5! 

25i4 

252O 

2526 

2532 

2538 

2544 

8 

52 

0286 

0293 

0299 

o3o6 

o3i2o3i8 

7 

52 

255i 

2557 

2563 

2569 

2575 

258i 

7 

53 

o325 

o33i 

o338 

o344 

o35i  o357 

6 

53 

2587 

2593 

2599 

26o5 

2611 

2617 

6 

54 

o364 

o37o 

o377 

o383 

o39o  o396 

5 

54 

2624 

263o 

2636 

2642 

2648 

2654 

5 

55 

o4o3 

0409 

o4i6 

0422 

o429  o435 

4 

55 

2660 

2666 

2672 

2678 

2684 

269o 

4 

56 

0442 

o448 

o454 

o46i 

o467  o474 

3 

56 

2696 

2702 

2709 

2715 

272I 

2727 

3 

57 

o48o 

o487 

o4g3 

o5oo 

o5o6  o5i3 

2 

57 

2733 

27392745 

2751 

2757 

2763 

2 

58 

o5i9 

o525 

0532 

o538 

o545o55i 

I 

58 

2769  2775:2781 

2787 

2793  2799 

I 

59 

o558 

o564 

0571 

o577 

o583'o59o 

O 

59 

28o5  2811:2817  2824 

283o2836  o 

60" 

50" 

40"  |  30"   20"  i  10" 

Cl 

60"     50"   40"   30"   20"   10"  )  fl. 

Co-sine  of  17  Degrees.     s 

Co-sine  of  16  Degrees. 

,,  p  .  (  1"  2"  3"  4"  5"  6"  7"  8"  9"   p  p  t  $  1"  2"  3"  4"  5"  6"  7"  8'  9" 
P-Part{  112334556   P'PartJ  112234456 

LOGARITHM    c    TANGENTS. 


.s 

Tangent  of  72  Degrees. 

£ 

Tangent  of  73  Degrees. 

aH 

0" 

10" 

20" 

30" 

40" 

50" 

S 

0" 

10" 

20" 

30" 

40''  1  50" 

o 

10.488224 

8296 

8367 

8439 

85n 

8582 

59 

o 

io.5i466i 

4736 

4812 

4887 

4962i5o38 

59 

I 

8654 

8726 

8797 

8869 

894i 

9013 

58 

I 

5ii3 

5i88 

5264 

5339 

54i5 

549o 

58 

2 

9o84 

9i56 

9228 

93oo 

937i 

9443 

57 

2 

5565 

564i 

57i6 

5792 

5867 

5943 

5  7 

3 

95i5 

9587 

9659 

973i 

9802 

9874 

56 

3 

6018 

6094 

6i69 

6245 

6320 

6396 

56 

4 

9946 

..18 

..90 

.162 

.234 

.3o6 

55 

4 

6471 

6547 

6623 

6698 

6774 

6849 

55 

5 

io.t,9o378 

o449 

0521 

o593 

o665 

o737 

54 

5 

69a5 

7ooi 

7o76 

7l52 

7228 

73o3 

54 

6 

o8o9 

0881 

o953 

1025 

io97 

1169 

53 

6 

7379 

7455 

753o 

76o6 

7682 

7758 

53 

7 

I24l 

i3i3 

i386 

i458 

i53o 

1602 

52 

7 

7833 

7909 

7985 

8061 

8137 

8212 

5a 

8 

1674 

i746 

1818 

i89o 

I962 

2o35 

5i 

8 

8288 

8364 

844o 

85i6 

8592 

8667 

5i 

9 

2107 

2I79 

225l 

2323 

2395 

2468 

5o 

9 

8743 

8819 

8895 

897i 

9o47 

9I23 

5o 

10 

io.49254o 

2612 

2684 

2757 

2829 

2901 

49 

:o 

io.5i9i99 

9275 

935i  94a7 

95o3 

9579 

49 

ii 

2973 

3o46 

3n8 

3i9o 

3263 

3335 

48 

ii 

9655 

9731 

98o7 

9883 

9959 

..35 

48 

12 

34o7 

348o 

3552 

3624 

3697 

3769 

47 

12 

IO.52OIII 

0187 

oa63 

o34o 

o4i6 

040,2 

47 

i3 

384i 

39i4 

3986 

4o59 

4i3i 

4204 

46 

i3 

o568 

o644 

o7po 

o797 

o873 

o949 

46 

i4 

4276 

4348 

442i 

4493 

4566 

4638 

45 

r4 

1025 

IIOI 

n78 

1254 

i33o 

1407 

45 

i5 

4711 

4783 

4856 

4928 

5ooi 

5074 

44 

i5 

i483 

i559 

i635 

I7I2 

i7»8 

1864 

44 

16 

5i46 

5219 

529I 

5364 

5437 

5509 

43 

16 

i94i 

2017 

2094 

2170 

2246 

2323 

43 

J7 

5582 

5654 

5727 

58oo 

5872 

5945 

42 

II 

2399 

2476 

2552 

2628 

27o5 

2781 

42 

18 

6018 
6454 

6090 
6527 

6i63 
6600 

6236 
6672 

63o9 
6745 

638i 
6818 

4i 
4o 

18 

2858 
33i7 

2934 
3394 

Son 
347o 

3o87 
3547 

3i64 
3623 

3700 

4i 
4o 

20 

10.496891 

6964 

7o36 

7I09 

-7182 

7255 

39 

20 

io.523777 

3853 

393o 

4007 

4o83 

4i6o 

o 

21 

7328 

74oi 

7474 

7547 

76i9 

6792 

38 

21 

4237 

43i3 

439o 

4467 

4544 

4620 

22 

7765 

7838 

7911 

7984 

8o57 

8i3o 

37 

22 

4697 

4774 

485i 

4927 

5oo4 

5o8i 

37 

23 

82o3 

8276 

8349 

8422 

8495 

8568 

36 

23 

5i58 

5235 

53i2 

5388 

5465 

5542 

36 

24 

864i 

87i4 

8787 

8860 

8934 

0007 

35 

24 

56i9 

5696 

5773 

585o 

5927 

6oo4 

35 

25 

9080 

9i53 

9226 

9299 

9372 

9445 

34 

25 

6081 

6i58 

6235 

63i2 

6389 

6466 

34 

26 

95i9 

9592 

9665 

9738 

9811 

9885 

33 

26 

6543 

6620 

6697 

6774 

685i 

6928 

33 

27 

9958 

.  io4 

.178 

.•Si 

.324 

32 

27 

7oo5 

-7082 

-7160 

7237 

73i4 

739i 

32 

28 

io.5oo397 

o47i 

o544 

0617 

o69i 

0764 

3i 

28 

7468 

7545 

7623 

77oo 

7777 

7854 

3i 

29 

o837 

0911 

0984 

1057 

n3i 

I2O4 

3o 

29 

793i 

8009 

8086 

8i63 

8241 

83i8 

3o 

3o 

10.501278 

i35i 

i425 

i498 

1571 

1645 

29 

3o 

io.528395 

8472 

855o 

8627 

87o5 

8782 

29 

3i 

I7i8 

1792 

i865 

i939 

2OI2 

2086 

28 

3i 

8859 

8937 

9oi4 

9091 

9i69 

924f5 

28 

32 

2l59 

2233 

2307 

238o 

2454 

2527 

27 

32 

9324 

94oi 

9479 

9556 

9634 

9711 

27 

3.1 

2601 

26-74 

2748 

2822 

2895 

2969 

26 

33 

9789 

9866 

9944 

.  .21 

••99 

.177 

26 

34 

3o43 

3n6 

3190 

3264 

3337 

34n 

25 

34 

io.53o254 

o332 

0409 

0487 

o565 

0642 

25 

35 

3485 

3559 

3632 

3706 

378o 

3854 

24 

35 

O720 

o798 

0875 

o953 

io3i 

1108 

24 

36 

892-7 

4ooi 

4o75 

4i49 

4223 

4296 

23 

36 

1186 

1264 

1  342 

i4i9 

i497 

1575 

23  ; 

37 

437o 

4444 

45i8 

4592 

4666 

474o 

22 

37 

i653 

i73i 

1808 

1886 

1964 

2042 

22 

38 

48i4 

4887 

4961 

5o35 

5  1  oo. 

5i83 

21 

38 

2I2O 

2I98 

2276 

2353 

243i 

2509 

21 

39 

5257 

533i 

54o5 

5479 

5553 

5627 

2O 

39 

2587 

2665 

2743 

2821 

2899 

3977 

2O 

4o 

10  ,5o57oi 

5775 

5849 

5923 

5997 

6o7i 

I9 

4o 

io.533o55 

3i33 

3an 

32b9 

3367 

3445 

I9 

4i 

6i46 

6220 

6294 

6368 

6442 

65i6 

18 

4i 

3523 

36o2 

368o 

3758 

3836 

39i4 

18 

42 

6590 

6664 

6739 

68i3 

6887 

696i 

*7 

42 

3992 

4o7o 

4i49 

4227 

43o5 

4383 

ll 

43 

7o35 

7no 

7i84 

7258 

7332 

7407 

16 

43 

446i 

454o 

46i8 

4696 

4774 

4853 

16 

44 

748i 

7555 

763o 

77o4 

7778 

7853 

i5 

44 

493i 

5oo9 

5o88 

5i66 

5244 

5323 

i5 

45 

7927 

8001 

8076 

8i5o 

8224 

8299 

i4 

45 

54oi 

5479 

5558 

5636 

57i5 

5793 

i4 

46 

8373 

8448 

8522 

8596 

867i 

8745 

i3 

46 

5872 

595o 

6028 

6107 

6i85 

6264 

i3 

47 

8820 

8894 

8969 

9o43 

9118 

9192 

12 

47 

6342 

6421 

6499 

6578 

6657 

6735 

12 

48 

9267 

934i 

94i6 

949o 

9565 

9640 

II 

48 

68i4 

6892 

6971 

7o5o 

7I28 

7207 

ll 

49 

97*4 

9-789 

9863 

9938 

.  .  i3 

..87 

10 

49 

7285 

7364 

7443 

75zz 

76oo 

7679 

IO 

5o 

io.5ioi62 

0237 

o3n 

o386 

o46i 

o535 

9 

5o 

io.537758 

7836 

79i5 

7994 

8o73 

8i5i 

9 

5i 

0610 

o685 

0760 

o834 

0909 

0984 

8 

5i 

823o 

83o98388 

8467 

8546 

8624 

8 

52 

1059 

ii34 

1208 

1283 

i358 

i433 

7 

52 

87o3 

8782)8861 

894o 

9oi9 

9o98 

7 

53 

i5o8 

1  582 

i657 

I732 

180-7 

1882 

6 

53 

9i77 

9256 

9335 

94i4 

9493 

9572 

6 

54 

i957 

2032 

2IO7 

2182 

2257 

2332 

*5 

54 

965i 

973o 

98o9 

9888 

9967 

..46 

5 

55 

24o7 

2482 

2557 

2632 

2707 

2782 

4 

55 

io.54oi25 

0204 

0283 

o362 

0442 

0521 

4 

56 

2857 

2932 

3oo7 

3082 

3i57 

3232 

3 

56 

0600 

o679 

o758 

0837 

o9i7 

o996 

3 

57 

33oy 

3382 

3457 

3533 

36o8 

3683 

2 

57 

io75 

ii54 

ia34 

i3i3 

i392 

l472 

2 

58 

3758 

3833 

39o8 

3984 

4069 

4i34 

I 

58 

i55i 

i63o 

I7IO 

1789 

1868 

i948 

I 

59 

4209 

4285 

436o 

4435 

45io 

4586 

O 

59 

2027 

2106 

2186 

2265 

2345 

2424 

O 

60" 

50"  |  40" 

30" 

20" 

10" 

C 

60" 

50" 

40" 

30" 

20" 

10" 

d 

Co-tangent  of  17  Degrees. 

Co-tangent  of  16  Degrees. 

a 

p  p   Cl"  2"  3"  4"  5"  6"  7"  8"  9" 

p  p   $  1"  2"  3"  4"  5"  6"  7"  8"  9" 

}  7  15  22  29  37  14  51  59  66 

I  8  15  23  31  39  46  54  62  70 

G 


LOGARITHMIC    SINES. 


.9 

Sine  of  74  Degrees. 

.3 

Sine  of  75  Degrees. 

i 

0" 

10" 

20"  1  30" 

40' 

50' 

» 

0" 

10" 

20"  |  30" 

40" 

50" 

o 

9.982842 

2848 

28542860 

2866 

2872 

59 

0 

9.984944 

4949 

4955 

496i 

4966 

4972 

69 

I 

2878 

2884 

2890  2896 

29O2 

2908 

58 

I 

4978 

4983 

4980 

4995 

5ooo 

5oo6 

58 

2 

2914 

292O 

2926  2932 

2938 

2944 

57 

2 

6011 

6017 

6023 

6028 

5o34 

5o4o 

5? 

o 

2960 

2956 

2962  2968 

3974 

2980 

56 

3 

5o45 

5o5i 

5o56 

6062 

5o68 

5o73 

56 

4 

2986 

2992  2998  3oo4 

3oio 

3oi6 

55 

4 

6079 

5o84 

6090 

5o96 

5ioi 

5io7 

55 

5 

3O22 

3028 

3o34|3o4o 

3o46 

3o52 

54 

5 

5n3 

5u8 

6124 

6129 

5i35 

5i4i 

54 

6 

3o58 

3o64 

3o7o  3o76 

3o82 

3o88 

53 

6 

5i46 

6162 

6167 

5i63 

5i69 

5i74 

53 

8 

3o94 
3i3o 

3ioo 
3i36 

3io6|3ii2 
3i4a3i48 

3n8 
3i54 

3i24 
3i6o 

52 

5i 

7 
8 

6180 
52i3 

5i85 
6219 

6191 

6224 

5i97 
523o 

6202 
5236 

6208 
6241 

62 
61 

9 

3i66 

3172 

3i783i84 

3i9o 

3i96 

5c 

9 

6247 

6262 

6268 

6264 

5269 

5275 

5o 

10 

5.983202 

3208 

32143220 

3226 

3232 

49 

IO 

5.986280 

6286 

6291 

6297 

53o3 

5368 

49 

ii 

3238 

3244 

325o3256 

3262 

3268 

48 

ii 

53i4 

53i9 

5325 

533o 

5336 

5342 

48 

12 

3273 

3279 

3285 

329i 

3297 

33o347 

12 

5347 

5353 

5358 

5364 

5369 

5375 

47 

i3 

SSog 

33i5 

332i 

3327 

3333 

3339 

46 

i3 

538i 

5386 

5392 

5397 

54o3 

54o8 

46 

i4 

3345 

335i 

3357 

3363 

3369 

3375 

45 

14 

54i4 

54i9 

6426 

543o 

5436 

5442 

45 

16 

338i 

3386 

3392 

3398 

34o4 

34io 

44 

16 

5447 

5453 

5458 

5464 

5469 

5475 

44 

16 

34i6 

3422 

3428 

3434 

344o 

3446 

43 

16 

548o 

5486 

6491 

5497 

6602 

55o8 

43 

17 

3452 

3458 

3464 

3469 

3475 

348i 

42 

»7 

55i4 

6619 

6626 

553o 

5536 

554i 

42 

18 

3487 

3493  3499 

35o5 

35n 

3617 

4i 

18 

5547 

6662 

5558 

5563 

5569 

5574 

4i 

19 

3523 

3529!3535 

354o 

3546 

3552 

4o 

19 

558o 

5585 

559i 

5596 

6602 

56o7 

4o 

20 

9.983668 

35643570 

3576 

3582 

3588 

39 

20 

9.986613 

56i8 

6624 

5629 

5635 

564o 

39 

21 

3594 

3599 

36o5 

36n 

36i7 

3623 

38 

21 

5646 

565i 

5657 

6662 

5668 

5673 

38 

22 

3629 

3635  364i 

3647 

3653 

3658 

37 

22 

6679 

5684 

6690 

5695 

57oi 

57o6 

3? 

23 

3664 

367o 

3676 

3682 

3688 

3694 

36 

23 

6712 

57i7 

5723 

6728 

5734 

5739 

36 

24 

3700 

3706  3711 

37i7 

3723 

3729 

35 

24 

5745 

575o 

6766 

6761 

5767 

5772 

35 

25 

3736 

374i 

3747 

3752 

3758 

3764 

34 

25 

5778 

5783 

6789 

5794 

6800 

58o5 

34 

26 

377o 

3776 

3782 

3788 

3794 

3799 

33 

26 

58xi 

58i6 

6822 

682? 

5832 

5838 

33 

27 

38o5 

38n 

38i7 

3823 

3829 

3835 

32 

27 

5843 

5849 

5854 

6860 

5865 

587i 

32 

28, 

384o 

3846 

3852 

3858 

3864 

3870 

3i 

28 

6876 

5882 

5887 

5893 

5898 

59o3 

3i 

29 

3876 

388i 

3887 

3893 

3899 

3906 

3o 

29 

6909 

6914 

6920 

5925 

593i 

5936 

3o 

3o 

9.983911 

39i6 

3922 

3928 

3934 

3940 

29 

3o 

9.986942 

5947 

5952 

5958 

5963 

5969 

29 

3i 

3946 

3961 

3957 

3963 

3969 

3975 

28 

3i 

6974 

6980 

5985 

599i 

5996 

6001 

28 

32 

3981 

3986  3992 

3998 

4oo4 

4oio 

27 

32 

6007 

6012 

6018 

6023 

6o29 

6o34 

27 

33 

4021 

4027 

4o33 

4o39 

4o45 

26 

33 

6o39 

6o45 

6060 

6066 

6061 

6o67 

26 

34 

4o5o 

4o56  4062 

4o68 

4o74 

4079 

25 

34 

6072 

6077 

6o83 

6088 

6o94 

6o99 

25 

35 

4o85 

4091 

4o97 

4io3 

4108 

4n4 

24 

35 

6io4 

6110 

6116 

6121 

6126 

6i32 

24 

36 

4l20 

4126  4i32 

4i37 

4i43 

4i49 

23 

36 

6137 

6142 

6i48 

6i53 

6i59 

6i64 

23 

37 

4i55 

4i6i 

4i66 

4l72 

4i784i84 

22 

37 

6169 

6176 

6180 

6186 

6i9i 

6i96 

22 

38 

4190 

4196 

4201 

4207 

421314218 

21 

38 

6202 

6207 

6212 

6218 

6223 

6229 

21 

39 

4224 

423o4236 

4242 

424-7 

4253 

2O 

39 

6234 

6239 

6246 

6260 

6266 

6261 

2O 

4o 

9.984269 

4266  4270 

4276 

4282 

4288 

19 

4o 

9.986266 

6272 

6277 

6282 

6288 

6293 

I9 

4i 

4294 

42994306 

43n 

43i7 

4322 

18 

4i 

6299 

63o4 

63o9 

63i5 

6320 

6325 

1  8 

42 

4328 

43344340 

4345 

435i 

4357 

17 

42 

633i 

6336 

6342 

6347 

6352 

6358 

17 

43 

4363 

4368  4374 

438o 

4386 

439i 

16 

43 

6363 

6368 

6374 

6379 

6384 

639o 

16 

44 
45 

4397 
4432 

44o344o9 
4437;4443 

44i4 
4449 

4420  4426 
445514460 

16 
i4 

44 
45 

6395 
6427 

64oi 
6433 

64o6 
6438 

64ii 
6443 

64i7 
6449 

6422 
6454 

16 

i4 

46 

4466 

4472  4477 

4483 

448oj4495 

i3 

46 

6459 

6465 

647o 

6475 

648  1 

6486 

i3 

47 

45oo 

45o645i2 

45i8 

4523I4529 

12 

47 

6491 

6497 

6602 

65o7 

65i3 

65i8 

12 

48 

4535 

454o  4546 

4662 

4558 

4563 

II 

48 

6523 

6629 

6534 

6539 

6545 

655o 

II 

49 

4569 

4575|458o 

4586 

4592 

4598 

IO 

49 

6555 

656i 

6566 

657i 

6577 

6682 

IO 

5o 

9.9846o3 

46094616 

4620 

4626 

4632 

9 

5o 

9.986687 

6693 

6598 

66o3 

6608 

66i4 

61 

4638 

4643  4649 

4655 

466o 

4666 

8 

61 

6619 

6624 

663o 

6635 

664o 

6646 

8 

62 

4672 

46774683 

4689 

4694 

4700 

7 

62 

6661 

6656 

6661 

6667 

66-72 

6677 

7 

53 

4706 

47i247i7 

4723 

4729 

4734 

6 

53 

6683 

6688 

6693 

6699 

67o4 

67o9 

6 

54 

474o 

4746475i 

4757 

4763j4768 

5 

54 

67i4 

6-720 

6725 

673o 

6736 

674i 

5 

55 

4774 

478o4785 

4791 

479748o2 

4 

55 

6746 

675i 

6757 

6762 

6767 

677.3 

4 

56 

48o8 

48i448iQ 

4826 

483i  4836 

3 

56 

6778 

6783 

6788 

6794 

6799 

68o4 

3 

57 

4842 

4848 

4853 

4859 

4865  487o 

2 

57 

6809 

6816 

6820 

6826 

683i 

6836 

2 

58 

4876 

4882 

4887 

4893 

4899  49o4 

I 

58 

684i 

6846 

6862 

6857 

686a 

6867 

I 

69 

4910 

4916 

492I 

4927 

49324938 

O 

59 

6873 

6878 

6883 

6888 

6894 

6899 

O 

60" 

50" 

40"   30" 

20"   10" 

. 

60" 

50" 

40" 

30" 

20" 

10" 

d 

Co-sine  of  15  Degrees. 

§ 

Co-sine  of  14  Degrees. 

S 

.  (  1"  2"  3' 

'  4"  5"  6"  7"  8"  9" 

p  p  A  I"  2"  3"  4"  5"  6"  7"  8"  <*" 

*l\  1   1   2 

233455 

P.Part^  1   i   o   2   3   3   4   4   ;> 

LOGARITHMIC    TANGENTS. 


1 

Tangent  of  74  Degrees. 

1 

Tangent  of  75  Degrees. 

s 

0" 

10" 

201 

30" 

40" 

50" 

S 

0" 

10" 

20" 

30" 

40" 

50" 

o 

io.54s5o^ 

2583 

2663 

2742 

2822 

29OI 

59 

O 

io.57i948 

2O32 

2116 

22OO 

2285 

2369 

59 

I 

2981 

3o6o 

3i4o 

3219 

3299 

3378 

58 

I 

2453 

2537 

2622 

2706 

2790 

2875 

58 

2 

3458 

3538 

3617 

3697 

3777 

3856 

57 

2 

2959 

3o44 

3i28 

3212 

3297 

338i 

57 

3 

3936 

4oi6 

4o95 

4i75 

4255 

4335 

56 

3 

3466 

355o 

3635 

37i9 

38o4 

3888 

56 

4 

44i4 

4494 

4574 

4654 

4733 

48i3 

55 

4 

3973 

4o58 

4i42 

4227 

43xi 

4396 

55 

5 

4893 

4973 

5o53 

5i33 

52i3 

5292 

54 

5 

448  1 

4565 

465o 

4735 

48i9 

4904 

54 

6 

5372 

5452 

5532 

56i2 

5692 

5772 

53 

6 

4989 

5o73 

5i58 

5243 

5328 

54i3 

53 

7 

5852 

5932 

6012 

6092 

6l72 

6252 

52 

7 

5497 

5582 

5667 

5752 

5837 

5922 

52 

8 

6332 

6412 

6492 

6572 

6653 

6733 

5i 

8 

6oo7 

6o9i 

6i76 

6261 

6346 

643i 

5i 

9 

68i3 

6893 

6973 

7o53 

7i33 

72l4 

5o 

9 

65i6 

6601 

6686 

677i 

6856 

6g4i 

5o 

10 

10.547294 

7374 

7454 

7535 

76i5 

7695 

49 

10 

io.577o26 

7II2 

7i97 

7282 

7367 

7452 

49 

ii 

7775 

7856 

7936 

8016 

8o97 

8177 

48 

ii 

?537 

7622 

7708 

7793 

7878 

7963 

48 

12 

8257 

8338 

84i8 

8498 

8579 

8659 

47 

12 

8o48 

8i34 

82I9 

83o4 

839o 

8475 

47 

i3 

874o 

8820 

8901 

8981 

9062 

9142 

46 

i3 

856o 

8646 

8731 

8816 

8902 

8987 

46 

i4 

9223 

93o3 

9384 

9464 

9545 

9625 

45 

i4 

9o73 

9i58 

9243 

9329 

94i4 

gSoo 

45 

i5 

9706 

9787 

9867 

9948 

,.28 

.  io9 

44 

i5 

9585 

967i 

9756 

9842 

9928 

..i3 

44 

16 

10.550190 

0270 

o35i 

o432 

o5i3 

o593 

43 

16 

10.580099 

0184 

0270 

o356 

0441 

0527 

43 

J7 

0674 

o755 

o836 

o9i6 

0997 

1078 

42 

*7 

o6i3 

0698 

0784 

0870 

0956 

io4i 

42 

18 

1  1  59 

I24O 

1320 

i4oi 

1482 

i563 

4i 

18 

1127 

I2l3 

1299 

i384 

1470 

i556 

4i 

'9 

1  644 

1725 

1806 

1887 

1968 

2049 

4o 

r9 

1642 

I728 

1814 

I9OO 

1986 

2072 

4o 

20 

io.552i3o 

2211 

2292 

2373 

2454 

2535 

39 

20 

io.582i58 

2243 

2329 

24i5 

2501 

2587 

39 

21 

2616 

2697 

2778 

a859 

2940 

3021 

38 

21 

2674 

276o 

2846 

2932 

3oi8 

3io4 

38 

22 

3l02 

3i83 

3265 

3346 

3427 

35o8 

37 

22 

3190 

3276 

3362 

3449 

3535 

362i 

37 

23 

3589 

367o 

3752 

3833 

39i4 

3995 

36 

23 

37o7 

3793 

388o 

3966 

4o52 

4i38 

36 

24 

4077 

4i58 

4239 

4321 

4402 

4483 

35 

24 

4225 

43n 

4397 

4484 

4570 

4657 

35 

25 

4565 

4646 

4728 

48o9 

489o 

4972 

34 

25 

4743 

4829 

4916 

5OO2 

5689 

5175 

34 

26 

5o53 

5i35 

52  1  6 

5298 

5379 

546i 

33 

26 

6262 

5348 

5435 

552i 

56o8 

5694 

33 

27 

5542 

5624 

0705 

5787 

5868 

595o 

32 

27 

578i 

5868 

5954 

6o4i 

6127 

6214 

32 

28 

6o32 

6n3 

6195 

6276 

6358 

644o 

3i 

28 

63oi 

6387 

6474 

656i 

6648 

6734 

3i 

29 

652i 

66o3 

6685 

6766 

6848 

693o 

3o 

29 

6821 

69o8 

6995 

7081 

7168 

7255 

3o 

3o 

10.557012 

7093 

7i75 

7257 

7339 

7421 

29 

3o 

io.587342 

7429 

75i6 

7603 

769o 

7776 

29 

Si 

75o3 

7584 

7666 

7748 

783o 

79I2 

28 

3i 

7863 

795o 

8o37 

8124 

82iij8298 

28 

32 

7994 

8076 

8i58 

8240 

8322 

84o4 

27 

32 

8385 

8472 

8559 

8647 

8734 

8821 

27 

33 

8486 

8568 

865o 

8732 

88i4 

8896 

26 

33 

8908 

8995 

9o82 

9i69 

9257 

9344 

26 

34 

8978 

9060 

9l42 

9224 

93o6 

9388 

25 

34 

943  1 

95i8 

96p5 

9693 

978o 

9867 

25 

35 

9471 

9553 

9635 

971? 

9799 

9881 

24 

35- 

9955 

..42 

.  I29 

.217 

.3o4 

.391 

24 

36 

9964 

..46 

.128 

.210 

.293 

.375 

23 

36 

10  590479 

o566 

o654 

0741 

o829 

0916 

23 

37 

io.56o457 

o54o 

0622 

0704 

0787 

0869 

22 

3? 

ioo4 

io9i 

1179 

1266 

i354 

i44i 

22 

38 

0952 

io34 

1116 

II99 

1281 

1  364 

21 

38 

1529 

1616 

1704 

I792 

i879 

1967 

21 

39 

i446 

1529 

1611 

1694 

1776 

i859 

20 

39 

2o55 

2l42 

2230 

23i8 

2406 

2493 

2O 

4o 

10.561941 

2024 

2106 

2189 

2272 

2354 

r9 

4o 

10.592581 

2669 

2757 

2845 

2932 

3020 

19 

4i 

2437 

2520 

2602 

2685 

2768 

285o 

18 

4i 

3io8 

3i96 

3284 

3372 

3460 

3548 

18 

42 

2933 

3oi6 

3099 

3i8i 

3264 

3347 

ll 

42 

3636 

3724 

38i2 

39oo 

3988 

4076 

17 

43 

343o 

35i2 

3595 

3678 

376i 

3844 

16 

43 

4i64 

4252 

434o 

4428 

45i6 

46o4 

16 

44 

3927 

4oio 

4093 

4i75 

4258 

434i 

i5 

44 

4692 

4781 

4869 

4957 

5o45 

5i33 

i5 

45 

4424 

45o7 

4590 

4673 

4756 

4839 

i.4 

45 

6222 

53io 

5398 

5486 

5575 

5663 

i4 

46 

4922 

5oo5 

5o88 

5172 

5a55 

5338 

i3 

46 

575i 

584o 

5928 

6oi7 

6io5 

6193 

i3 

47 

542i 

55o4 

5587 

5670 

5754 

5837 

12 

47 

6282 

637o 

6459 

6547 

6636 

6724 

12 

48 

5920 

6oo3 

6086 

6170 

6253 

6336 

I  I 

48 

68i3 

69oi 

699o 

7o78 

7i67 

7256 

II 

49 

6420 

65o3 

6586 

6669 

6753 

6836 

10 

49 

7344 

7433 

7522 

76io 

7699 

7788 

10 

5o 

10.566920 

7003 

7086 

7170 

7253 

7337 

9 

5o 

10.597876 

7965 

8o54 

8i43 

823i 

8320 

9 

5i 

7420 

75o4 

7587 

7671 

7754 

7838 

8 

5i 

8409 

8498 

8587 

8675 

8764 

8853 

8 

52 

7921 

8oo5 

8088 

8172 

8255 

8339 

7 

52 

8942 

9o3i 

9120 

92O9 

9298 

9387 

7 

53 

8423 

85o6 

859o 

8674 

8757 

884i 

6 

53 

9476 

9565 

9654 

9743 

9832 

992I 

6 

54 

8925 

9008 

9092 

9176 

9260 

9343 

5 

54 

10.600010 

OIOO 

oi89 

0278 

o367 

o456 

5 

55 

9427 

95n 

9595 

9679 

9763 

9846 

4 

55 

o545 

o635 

0724 

o8i3 

O9O2 

O992 

4 

56 

993o 

..i4 

..98 

.182 

.266 

.35o 

3 

56 

1081 

II70 

1260 

1  349 

i438 

i528 

3 

57 

10.570434 

o5i8 

0602 

0686 

o77o 

o854 

2 

57 

1617 

I7o6 

1796 

i885 

i975 

2064 

2 

58 

og38 

1022 

1  1  06 

1190 

I274 

i358 

I 

58 

2i54 

2243 

2333 

2422 

25l2 

2601 

I 

59 

1442 

1527 

1611 

i695 

I779 

i863 

O 

59 

269I 

278l 

2870 

296o 

3o5o 

3i39 

0 

60"      50" 

40" 

30" 

20" 

10" 

~ 

60"     |  50" 

40" 

30" 

20" 

10" 

. 

fVtanger.t  of  15  Degrees. 

§ 

Co-tangent  of  14  Degrees. 

1 

P  Pa  t$  !"   2"   3"   4"   5"  6"   7"   8"   9" 

lrt$  8  If  25  33  41  49  57  65  74 

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00 


L  0  G  A  R  IT  H  M  I  C 


N  E  S. 


I 

Sine  of  76  Degrees. 

.5 

Sine  of  77  Decrees. 

si 

0" 

10"  |  20" 

30" 

40" 

50" 

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0" 

10" 

20" 

30" 

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50" 

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60" 

50" 

40" 

30"  I  20" 

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0 

60" 

50" 

40" 

30" 

20" 

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d 

Co-sine  of  13  Degrees. 

2 

Co-sine  of  12.  Degrees. 

1 

p//  o"  3"  4"  5"  6"  7//  8"  9" 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

irtj  1   1   2   2   3   3   4   4   5 

P.  Part  ^  011223344 

L,  O  f;  A  R  I  T  H  Al  I  C      T  A  N  fi  F.  N  T  3. 


r, 

Tangent  of  76  Degrees. 

d 

Tangent  of  77  Degrees. 

1  a 

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22l5 

23i8 

2422 

O 

60" 

50" 

40" 

30" 

20" 

10" 

j 

60"    |  50"   40"  |  30"  |  20" 

10" 

d 

— 

Co-tangent  of  13  Degrees. 

^_ 

Co-tangent  of  12  Degrees. 

•i 

P  Part 
irt 


8"    9' 


9     19    28    37    46    56    65    74    83 


I    P.  Part 


3"    4"    5"    6"    7"    8"    9' 
10    20*  30    40    50    6T)    70    80    90 


02 


LOGARITHMIC    IS  i  N  E  &. 


J 

Sine  of  78  Degrees. 

d 

;r! 

Sine  of  79  Degrees. 

* 

0" 

10"  |  20"   30"   40" 

50" 

2 

0" 

10" 

20" 

30" 

40'' 

50" 

o 

9.99o4o4 

o4og  o4i3  o4i8  0422 

042  7 

69 

o 

9.99i947 

i95i 

i95b 

i95g 

i963 

i967 

59 

I 

o43i 

o436  o44o  o445 

o449 

o454 

58 

i 

i97i 

10.75 

I979 

I983 

i987 

I992 

58 

2 

o458 

0462  0467  o47i 

o4?6 

0480 

57 

2 

I996 

2OOO 

2008 

2OI2 

2016 

57 

3 

o485 

0489  0494  0498 

o5o3 

o5o7 

56 

3 

2O2O 

2O24 

2028 

2032 

2o36 

2040 

56 

4 

o5n 

o5i6  o52o  o525 

0529 

o534 

55 

4 

2044 

2049 

2o53 

2057 

2061 

2o65 

55 

5 

o538 

o543  o547  o552 

o556 

o56o 

54 

5 

2o69 

2073 

2077 

2081 

2o85 

2o89 

54 

6 

o565 

o569jo574  0578 

o583 

o587 

53 

6 

2O93 

2O97 

2IOI 

2105 

2IO9 

2Il3 

53 

7 

©591 

o596 

0600  o6o5 

0609 

0614 

52 

7 

2118 

2122 

2126 

2130 

2i34 

2i38 

52 

,  8 

0618 

0622 

o627 

o63i 

o636 

o64o 

5i 

8 

2l42 

2l46 

2i5o 

2i54 

2i58 

2162 

5i 

9 

o645 

0649 

o653 

o658 

0662 

0667 

5o 

9 

2l66 

2I70 

2I74 

2I78 

2182 

2186 

00 

10 

9.99o67i 

o675 

0680 

0684 

0689 

0693 

49 

10 

9.992I90 

2I94 

2I98 

22O2 

2206 

2210 

49 

ii 

o697 

0702 

0-706 

07II 

0715 

0719 

48 

ii 

22l4 

22l8 

2222 

2226 

2230 

2234 

48 

12 

0724 

0728 

o733 

o737 

0741 

0746 

47 

12 

2239 

2243 

2247 

225l 

2255 

225g 

47 

i3 

o75o 

o755 

o759 

o763 

0768 

0772 

46 

i3 

2263 

2267 

227I 

2275 

2279 

2283 

46 

i4 

0777 

0781 

o785 

o79o 

0794 

0798 

45 

i4 

2287 

229I 

2295 

2299 

23o3 

23o7 

45 

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o8o3 

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0812 

0816 

0820 

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44 

i5 

23ll 

23i5 

23l9 

2323 

2327 

233i 

44 

16 

0829 

o833 

o838 

0842 

0847 

o85i 

43 

16 

2335 

233q 

2343 

2347 

235i 

2355 

43 

i7 

o855 

0860 

08640868 

o873 

0877 

42 

17 

2359 

2363 

2366 

2370 

2374 

2378 

42 

18 

0882 

0886 

0890 

o895 

0899 

0903 

4i 

18 

2382 

2386 

239o 

2394 

2398 

2402 

4i 

20 
21 
22 

0908 
9.990934 
0960 
0986 

0912 
o938 
0964 
0990 

0916 
0942 
0969 
o995 

0921 

0947 
o973 

°999 

0925 
0951 

°977 
ioo3 

0929 
o955 
0982 
1008 

4o 
39 

38 
37 

20 

21 
22 

2406 

9.992430 
2454 
2478 

2410 

-2434 
2458 
2482 

24i4 
2438 
2462 
2485 

2418 
2442 
2466 
248g 

2422 

2446 

2470 

2493 

2426 

2474 

2497 

4o 
39 
38 
37 

23 

IOI2 

1016,1021 

IO25 

1029 

io33 

36 

23 

25oi 

25o5 

25og 

25i3 

25l7 

2521 

36 

24 

io38 

1042  1046 

io5i 

io55 

io5g 

35 

24 

2525 

2529 

2533 

2537 

2541 

2545 

35 

25 

io64 

1068  1072 

io77 

1081 

io85 

34 

25 

2549 

2553 

2556 

256o 

2564 

2568 

34 

26 

1090 

1094  1098 

no3 

IIO7 

mi 

33 

26 

2572 

2576 

258o 

2584 

2588 

2592 

33 

27 

iiiS 

1120,1124 

1128 

n33 

1137 

32 

27 

2596 

2600 

2604 

26o7 

2611 

26i5 

32 

28 

n4i 

n46  n5o 

n54 

n58 

n63 

3i 

28 

2619 

2623 

2627 

263i 

2635 

263g 

3i 

29 

1167 

1171  1176 

1  1  80 

n84 

1188 

3o 

29 

2643 

2647 

265i 

2654 

2658 

2662 

3o 

3o 

9.991193 

1197 

1201 

1206 

1210 

I2l4 

29 

3o 

9.992666 

2670 

2674 

2678 

2682 

2686 

29 

3i 

1218 

1223 

I227 

I23l 

1235 

1240 

28 

3i 

2690 

2693 

2697 

27OI 

27o5 

2709 

28 

32 

1244 

1248  1253 

I257 

1261 

1265 

27 

32 

2713 

27I7 

272I 

2725 

2728 

2732 

27 

33 

1270 

1274  1278 

1282 

I287 

1291 

26 

33 

2736 

2744 

2748 

2752 

2756 

26 

34 

1295 

1299  i3o4 

i3o8 

I3l2 

i3i6 

25 

34 

2759 

2763 

2767 

277I 

2775 

2779 

25 

35 

l32I 

i325  1329 

i333 

i338 

1  342 

24 

35 

2783 

2787 

2790 

2794 

2798 

2802 

24 

36 

1  346 

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1  35  9 

i363 

1367 

23 

36 

2806 

2810 

28l4 

28l8 

2821 

2825 

23 

37 

1372 

i376  i38o 

1  384 

1389 

i393 

22 

37 

2829 

2833 

2837 

284l 

2845 

2848 

22 

38 

i397 

i4oi  i4o6 

i4io 

i4i4 

i4i8 

21 

38 

2852 

2856 

2860 

2864 

2868 

287I 

21  * 

39 

1422 

i427ii43i 

i435 

i439 

i444 

2O 

39 

2875 

2879 

2883 

2887 

289I 

2895 

2O 

4o 

9.99i448 

i452  i456 

i46o 

i465 

1469 

19 

4o 

9.992898 

29O2 

29o6 

29IO 

29i4 

29l8 

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4i 

i473 

i477  1482 

i486 

i49o 

i494 

18 

4i 

2921 

2925 

292Q 

2933 

2937 

294i 

18 

i498 

i5o3  i5o7 

i5n 

i5i5  i5i9 

17 

42 

2944 

2948 

2952 

2956 

296o 

2963 

i7 

43 

i524 

i528 

1532 

i536 

i54oi545 

16 

43 

2967 

297I 

2975 

2979 

2983 

2986 

16  - 

44 

i549 

i553 

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i56i 

i566  i57o 

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44 

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2994 

2998 

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3oo5 

3oo9 

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45 

i574 

i578 

i582 

i586 

i59i  i595 

i4 

45 

3oi3 

3oi7 

3O2I 

3024 

3028 

3o3s 

i4 

46 

i599 

i6o3 

i6o7 

1612 

1616  1620 

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46 

3o36 

3o4o 

3o43 

3o47 

3o5i 

3o55 

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47 

1624 

1628 

i632 

i637 

i64i  i645 

12 

47 

3o59 

3o62 

3o66 

3o7o 

3o74 

3o78 

12 

48 

i649 

i653 

i657 

1662 

1666  i67o 

II 

48 

3o8i 

3o85 

3o89 

3o93 

3o97 

3ioo 

II 

49 

1674 

i678 

1682 

i687 

169111695 

10 

49 

3io4 

3io8 

3lI2 

3n5 

3n9 

3i23 

IO 

5o 

9.991699 

I7o3 

I7O7 

I7I2 

1716  1720 

9 

5o 

9.993i27 

3i3i 

3i34 

3i38 

3l42 

3i46 

9 

5i 

1724 

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I732 

i736 

i74i  i745 

8 

5i 

3i49 

3i53 

3i57 

3i6i 

3i65 

3i68 

8 

52 

1749 

i753 

i757 

1761 

i765  i77o 

7 

52 

3172 

3i76 

3i8o 

3i83 

3i87 

3i9i 

7 

53 

1774 

i778 

I782 

1786 

i79o  i794 

6 

53 

3i95 

3i98 

32O2 

32o6 

3210 

32i3 

6 

54 

1799 

i8o3 

i8o7 

1811 

1816  1819 

5 

54 

3217 

3221 

3225 

3228 

3232 

3236 

5 

55 

182! 

1827 

i832 

i836 

i84o  1  844 

4 

55 

324o 

3243  3247 

325i 

3255 

3258 

4 

56 

1  848 

i852 

i856 

1860 

18661869 

3 

56 

3262 

3266  327O 

3273 

3277 

3281 

3 

57 

i873 

i877 

1881 

i885 

1889  1893 

2 

57 

3284 

3288 

3292  3296 

3299  33o3 

2 

58 

1897 

1901 

1006 

IQIO 

1914  1918 

I 

58 

3307 

33n 

33i4  33i8  3323,3325 

I 

59 

1922 

1926  1930  1934 

1938  1942 

O 

59 

3329 

33333337334o3344!3348 

O 

60" 

50"   40"   30"  1  20"   10" 

fl 

60"     50"   40"  i  30"   SO"   10" 

j 

Co-sine  of  1  1  Degrees. 

1 

Co-sine  of  10  Degrees. 

* 

P  Part  5  1/;   2"   3"   4"   5"   6"  7"   8//  9" 

r'rari)  0*1   1223334 

C  l"  2"  3"  4"  5"  6"  7"  S"  9" 
f.  rart  <n   11222334 

•                          I       v                          J 

LOGARITHMIC    TANGENTS. 


103 


1 

Tangent  of  78  Degrees. 

.3 

Tangent  of  79  Degrees. 

«•! 

0" 

10" 

20" 

30" 

40" 

50" 

3 

0" 

10" 

20" 

30" 

40" 

50" 

o 

10.672525 

2629. 

2733 

2836 

294o 

3o43 

59 

0 

10.711348 

i46o 

i573 

i685 

I798 

I9IO 

59 

I 

3i47 

325i 

3354 

3458 

3562 

3666 

58 

i 

2023 

2i35 

2248 

236i 

2473 

2586 

58 

2 

3769 

3873 

3977 

4o8i 

4i85 

4289 

57 

2 

2699 

2811 

2924 

3o37 

3i5o 

3263 

57 

3 

4393 

44g7 

46oi 

47o5 

48o9 

49i3 

56 

3 

3376 

3488 

36oi 

37i4 

3827 

394o 

56 

4 

5017 

5l2I 

5225 

5329 

5433 

5537 

55 

4 

4o53 

4i67 

4280 

4393 

45o6 

46i9 

55 

5 

5642 

5746 

585o 

5954 

6o59 

6i63 

54 

5 

4732 

4846 

4959 

5072 

5i85 

5299 

54 

6 

6267 

6372 

6476 

658o 

6685 

6789 

53 

6 

54i2 

5526 

5639 

5752 

5866 

5979 

53 

7 

6894 

6998 

7io3 

7207 

7312 

7417 

52 

7 

6o93 

6207 

6320 

6434 

6547 

6661 

52 

8 

7521 

7626 

773i 

7835 

794o 

8o45 

5i 

8 

6775 

6889 

7002 

7116 

723o 

7344 

5i 

9 

8149 

8254 

8359 

8464 

8569 

8674 

5o 

9 

7458 

7572 

7686 

7799 

79i3 

802-7 

5o 

10 

8778 

8883 

8988 

9093 

9i98 

93o3 

49 

10 

8142 

8256 

8370 

8484 

8598 

87I2 

49 

ii 

94o8 

95i3 

96i8 

9723 

9829 

9934 

48 

ii 

8826 

8941 

9o55 

9169 

9283 

9398 

48 

12 

10  680039 

oi44 

0249 

o355 

o46o 

o565 

47 

12 

95l2 

9627 

974i 

9856 

997° 

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47 

i3 

0670 

0776 

0881 

0987 

IO92 

1197 

46 

i3 

I0.720I99 

o3i4 

0428 

o543 

o658 

0772 

46 

i4 

i3o3 

1408 

i5i4 

1619 

1725 

i83o 

45 

i4 

o887 

1002 

1116 

I23l 

1  346 

i46i 

45 

i5 

1936 

2042 

2147 

2253 

2359 

2464 

44 

i5 

i576 

1691 

1806 

1921 

2o36 

2l5l 

44 

16 

2570 

2676 

2782 

2887 

2993 

3099 

43 

16 

2266 

238i 

2496 

2611 

2726 

2841 

43 

17 

32o5 

33n 

3417 

3523 

3629 

3735 

42 

17 

2957 

3072 

3i87 

3302 

34i8 

3533 

42 

18 

384i 

3947 

4o53 

4i59 

4265 

437i 

4i 

18 

3649 

3764 

3995 

4no 

4226 

4i 

"9 

4477 

4584 

4690 

4796 

4902 

5009 

4o 

19 

4342 

4457 

4573 

4688 

48o4 

4920 

4o 

20 

5n5 

5221 

5328 

5434 

554o 

5647 

39 

20 

5o36 

5i5i 

5267 

5383 

5499 

56i5 

39 

21 

5753 

586o 

5966 

6073 

6i79 

6286 

38 

21 

573i 

5847 

5963 

6o79 

6i95 

63n 

38 

22 

6392 

6499 

6606 

6712 

6819 

6926 

37 

22 

6427 

6543 

6659 

6775 

689i 

7oo8 

37 

23 

7032 

7i39 

7246 

7353 

7460 

7567 

36 

23 

7124 

7240 

7356 

7473 

7589 

77o6 

36 

24 

7673 

7780 

7887 

7994 

8101 

8208 

35 

24 

•7822 

7938 

8o55 

8i7i 

8288 

84o5 

35 

r 

83i5 

8422 

8529 

8636 

8744 

885i 

34 

25 

852i 

8638 

8754 

887i 

8988 

9io5 

34 

26 

8958 

9o65 

9I72 

9280 

9387 

9494 

33 

26 

9221 

9338 

9455 

9572 

9689 

9806 

33 

27 

9601 

9709 

98i6 

9924 

.i38 

32 

27 

9923 

..4o 

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.5o8 

32 

38 

10.690246 

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28 

io.73o625 

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0860 

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I2II 

3: 

29 

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1107 

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3o 

29 

1329 

1  446 

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3o 

3o 

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1  645 

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1861 

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2077 

29 

3o 

2033 

2161 

2268 

2386 

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2621 

29 

3i 

2i84 

2292 

2400 

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2724 

28 

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2739 

2856 

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32IO 

3327 

28 

32 

2832 

294i 

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3373 

27 

32 

3445 

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27 

33 

348  1 

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3698 

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26 

33 

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34 

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42  39 

4348 

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25 

34 

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4980 

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5335 

5453 

25 

35 

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22 

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21 

39 

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783o 

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39 

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8660 

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20 

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19 

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18 

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18 

42 

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947i 

958i 

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io5o 

1170 

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43 

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1118 

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44 

2010 

2i3o 

2250 

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1668 

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45 

273l 

285i 

297I 

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3212 

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46 

1999 

2109 

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2440 

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46 

3453 

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47 

2661 

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2992 

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12 

47 

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12 

48 

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48 

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49 

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10 

49 

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6352 

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6716 

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9 

5i 

53i6 

5427 

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576i 

5872 

8 

5i 

7080* 

7201 

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7566 

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8 

52 

5983 

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52 

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53 

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6761 

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54 

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2 

57 

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2 

58 

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60" 

50" 

40" 

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e 

60" 

50" 

40" 

30" 

20" 

10" 

rt 

Co-tangent  of  1  1  Degrees. 

•fl 

Co-tangent  of  10  Degrees. 

i 

p  p  .  $  1"  2"  3"  •  4"  5"  6"  1"  8'1  9" 

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04 


LOGARITHMIC    SINE&. 


1 

Sine  of  80  Degrees. 

,c 

Sine  of  81  Degrees. 

m 

0" 

10" 

20" 

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60" 

50" 

40" 

30"   20" 

10" 

d 

60"    |  50"   40"   30"   20"   10"   ^ 

Co-sine  of  9  Degrees. 

•5 

s 

Co-sine  of  8  Degrees.      a 

P  Part*  l"  2"  3"  4"  5//  6//  7//  8//  9" 
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10? 


e     Taiurent  of  80  Degrees. 

d 

Tangent  of  81  Degrees. 

i  a;   & 

10"  |  20" 

30" 

40" 

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7977 

7299 
8112 

7435 

8248 

757o77o6 

838485i9 

3 

2 

56 

57 

8546 
9456 

869-78849 
9608  9-760 

9001 
9912 

9152  9304 
..64!.  216 

3 

2 

58 
59 

8655 
9471 

8791189279063191999335 
96o7'9743  9879  .  .  i5  .  i5i 

I 
0 

58 
59 

io.85o368  05200672 

1282  i434'i587 

0825 
i739 

o977|ii29 

l892  2045 

I 
0 

60"      50" 

40"  |  30"   20"   10" 

d 

60"      50"   40" 

30" 

20"  I  10" 

.: 

Co-tangent  of  9  Degrees 

.3 

& 

Co-tangent  of  8  Degrees. 

2 

P  P«  t$  !"  2//  3//  4"  5"  6//  7//  8;'  9' 
Lrl}  13  26  39  52  65  78  91  103  116 

P  Part}  r/  2"  3"  4"  5"  G"  7"  8"  9"  1 
llt)  14^9  43  58  72  86  101  115  130  1 

00 


LOGARITHMIC    SINES. 


c      Sine  of  82  Degrees. 

d 

Sine  of  83  Degrees. 

* 

0" 

10" 

20" 

30" 

40" 

50" 

s 

0" 

10" 

20" 

30" 

40" 

50" 

o 
i 

9.995753 
577i 

5756 
5773 

5759 
5776 

5762 
5779 

5765 
5782 

5768 
5785 

59 

58 

o 

9.996751 
6766 

6753 
6769 

6756 
677i 

6758 
6774 

676i 
6777 

6764 
6779 

59 
58 

2 

3 

5788 
58o6 

579i 
58o9 

58i2 

5797 

58oo 
58i8 

58o3 
582i 

57 
56 

2 

6782 
6797 

6784 
6800 

6787 
6802 

6789 
68o5 

6792 
68o7 

6795 
6810 

57 
56 

4 

5823 

5826 

5829 

5832 

5835 

5838 

55 

L 

6812 

68i5 

6818 

6820 

6823 

6825 

55 

5 

584i 

5844 

5847 

585o 

5853 

5856 

54 

£ 

6828 

683o 

6833 

6835 

6838 

54 

6 

5859 

5862 

5864 

5867 

587o 

5873 

53 

6 

6843 

6846 

6848 

685i 

6853 

6856 

UfJ. 

53 

7 

5876 

5879 

5882 

5885 

5888 

5891 

52 

7 

6858 

6861 

6863 

6866 

6869 

6871 

52 

8 

5894 

5897 

5899 

5902 

59o5 

59o8 

5i 

8 

68  74 

6876 

6879 

6881 

6884 

/ 

6886 

5i 

9 

10 

ii 

5911 

9.995928 
5946 

59i4 
593i 
5949 

59i7 
5934 
5g52 

5920 
5937 
5954 

5923 
5940 
5957 

5926 
5943 
5960 

5o 

49 

48 

9 

10 

ii 

6889 
9.996904 
6919 

689i 
69o6 

6922 

6894 
69°9 

6896 

69I2 

6899 
6914 
6929 

6901 
69i7 
6932 

5o 

49 

12 

5963 

f   r\ 

59bb 

59b9 

5972 

6976 

6978 

47 

12 

6934 

6937 

6939 

6942 

6944 

6947 

47 

i3 

598o 

5983 

5986 

5989 

5992 

5995 

46 

i3 

6949 

6952 

6954 

6957 

6959 

6962 

46 

i4 
i5 
16 

5998 
6oi5 
6o32 

6001 
6018 
6o35 

6oo3 
6021 
6o38 

6006 
6o23 
6o4i 

6oo9 
6026 
6o43 

6012 
6029 
6o46 

45 
44 
43 

i4 
i5 
16 

6964 
6979 
6994 

6967 
6982 
6997 

6969 

6984 

6999 

6972 

6987 

7OO2 

6974 
6989 
7004 

6977 
6992 
7oo7 

45 
44 
43 

ll 

6o49 

6o52 

6o55boi>8 

6061 

6o63 

42 

17 

7009 

7OII 

7014 

7oi6 

7oi9 

7021 

42 

18 

6066 

6o69 

6o72  6o75 

6o78  6081 

4i 

18 

7024 

7026 

7020. 

7o3i 

7o34 

7o36 

4i 

i9 

6o83 

6086 

6089  6092 

6o95|6o98 

4o 

J9 

7o39 

7o4i 

7044 

7o46 

7o5i 

4o 

20 

o  996ioo 

6io3 

6106  6109 

6112  6n5 

39 

20 

9.997053 

7o56 

7o58 

7o6i 

7o63 

7066 

39 

21 

6n7 

6120 

6i23  6126 

6l29 

6i3i 

38 

21 

7068 

7o7i 

7073 

7o76 

7o78 

7080 

38 

22 
23 
24 

6i34 
6i5i 
6168 

6i37 
6i54 
6i7i 

6i4o6i43 
6i57  6160 
6174:6177 

6i46 
6162 
6i79 

6i48 
6i65 
6182 

36 
35 

22 
23 

24 

7o83 
7o98 

7II2 

7o85 

7IOO 

7n5 

7088 
7102 
7117 

7o9o 
7io5 

7I2O 

7o93 
7I07 

7I22 

7095 
7110 
7124 

3? 
36 
35 

25    6i85 

6188 

6191,6193 

6196 

6199 

34 

25 

71-27 

7I29 

7i34 

7i37 

7i39 

34 

26 

6202 

62o5 

6207  6210 

62136216 

33 

26 

7i4i 

7i44 

7i46 

7i4g 

7i5i 

33 

27 

62I9 

6221 

6224  6227 

623o  6232 

32 

27 

7i56 

7i58 

7161  7163 

7i66 

7168 

32 

28 

6235 

6238 

6241  6244 

6246 

6249 

3i 

28 

7i7o 

7i73 

7i75 

7i78 

7i8o 

7182 

3i 

29 

6252 

6255 

6257  6260 

6263 

6266 

3o 

29 

7i85 

7i87 

7i9o 

•7192 

7i94 

7197 

So 

3o 

Q  >  996269 

627I 

6274  6277 

6280 

6282 

29 

3o 

9-997J99 

72O2 

7204 

7ao6 

72O9 

7211 

29 

3i 

6285 

6288 

6291  6293 

6296 

6299 

28 

3i 

7214 

72l6 

7218 

722I 

7223 

7226 

28 

32 

63o2 

63o5 

63076310 

63i3 

63i6 

27 

32 

*  7228 

723o 

7233 

7235 

7238 

7240 

27 

33 

63i8 

632i 

63246327 

6329 

6332 

26 

33 

7242 

7245 

7247 

7249 

7252 

26 

34 

6335 

6338 

634o6343 

6346 

6349 

25 

34 

7257 

7259 

7261 

7264 

7266 

7268 

25 

35 

635i 

6354 

63576359 

6362 

6365 

24 

35 

727I 

7273 

7276 

7278 

7280 

7283 

24 

36 

6368 

637o 

63736376 

6379 

638i 

23 

36 

7285 

7287 

729o 

7292 

7294 

7297 

23 

37 

6384 

6387 

6390  6392 

6395 

6398 

22 

37 

7299 

73oi 

73o4 

73o6 

73o9 

73n 

22 

?,8 

64oo 

64o3 

64o6  6409 

64n 

64i4 

21 

38 

73i3 

73i6 

73i8 

7320 

7323 

7325 

21 

89 

64i7 

64i9 

6422  6425 

6428 

643o 

20 

39 

7327 

733o 

7332 

7334 

7337 

7339 

2O 

4o 

9  996433 

6436 

6438644i 

6444 

6447 

9 

4o 

7344 

7346 

7348 

735i 

7353 

I9 

4i 

6449 

6452 

64556457 

646o 

6463 

8 

4i 

7355 

7358 

736o 

7362 

7365 

7367 

18 

42 

6465 

6468 

6471  6474 

6476 

6479 

7 

42 

7369 

7372 

7374 

7376 

7379 

738i 

17 

43 

6482 

6484 

6487  6490 

6492 

6495 

6 

43 

7383 

7386 

7388 

739° 

7393 

7395 

16 

44 

6498 

65oo 

65o3  65o6 

65o8 

65n 

5 

44 

7397 

7399 

7402 

74o4 

74o6 

74o9 

i5 

45 

65i4 

65i7 

55l9  6522 

6525 

6527 

4 

45 

74n 

74i3 

74i6 

74i8 

742O 

7423 

i4 

46 

653o 

6533 

6535  6538 

654i 

6543 

3 

46 

7425 

7427 

7429 

7432 

7434 

7436 

i3 

47 

6546 

6549 

655i  6554 

6557 

6559 

2 

47 

7439 

744i 

7443 

7445 

7448 

745o 

12 

48 

6562 

6565 

65676570 

6573 

6575 

I 

48 

7452 

7455 

7457 

7459 

746i 

7464 

II 

49 

6578 

658o 

6583 

6586 

6588 

659i 

O 

49 

7466 

7468 

7471 

7473 

7475 

7477 

IO 

5o 

9.  996594 

6596 

6599 

6602 

66o4 

66o7 

9 

5o 

9.997480 

7482 

7484 

7487 

7489 

749i 

9 

5i 

6610 

6612 

66i5 

6618 

6620 

6623 

8 

5i 

7493 

7496 

7498 

75oo 

75O2 

75o5 

8 

52 

6625 

6628 

663i 

6633 

6636 

6639 

7 

52 

75o7 

75o9 

75n 

75i4 

75i6 

75i8 

7 

53 

664i 

6644 

6646 

6649 

6652 

6654 

6 

53 

752O 

7523 

7525 

7527 

753o 

7532 

6 

54 

6657 

6660 

6662 

6665 

6667 

667o 

5 

54 

7534 

7536 

7539 

754i 

7543 

7545 

5 

55 

6673 

6675 

6678 

6681 

6683 

6686 

4 

55 

7547 

755o 

7  552 

7554 

7556 

7559 

4 

56 

6688 

669i 

6694 

6696 

6699 

67oi 

3 

56 

756i 

7563 

7565 

7568 

757o 

7572 

3 

57 

67o4 

67o7 

670916712  6714 

67i7 

2 

57 

7574 

7577 

7579 

758i 

7583 

7585 

2 

58 

6720 

6722 

6725  6727  6730 

6733 

I 

58 

7588  759o 

7592 

7594 

7597 

7599 

I 

59 

6735 

6738 

6740 

6743  6746 

6748 

0 

59 

76oi|76o3 

76o5 

76o8  76io  76i2  o 

6CX' 

50" 

40" 

30"   20" 

10" 

. 

60"    |  50"  [  40"   30"   20"   10"  |  ^ 

Co-sine  of  7  Degrees. 

a* 

Co-sine  of  6  Degrees.      § 

<  1"  2"  3   4'  V  6"  7"  8"  9" 

p  p  .  (  1"  2"  3"  4"  5"  6"  7"  8   9"  n 

irt$  0   1   1   1   12222 

irt$  0   0   1   1   11222 

LOGARITHMIC    TANGENTS. 


101 


a 

Tangent  of  82  Degrees. 

| 

Tangent  of  83  Degrees. 

§ 

0" 

10" 

20" 

30" 

40" 

50" 

2 

0" 

10" 

20" 

30" 

40" 

50" 

o 

10.  852197 

235o 

25o3 

2656 

28o9 

29.62 

59 

0 

io.9io856 

io3o 

1205 

i379 

i553 

1727 

59 

I 

3:  1  5 

3268 

3421 

3575 

3728 

388i 

58 

i 

I9O2 

2076 

225l 

2426 

2600 

2775 

58 

2 

4o34 

4i88 

4341 

4495 

4649 

4802 

57 

2 

295o 

3i25 

33oo 

3475 

365o 

3825 

57 

3 

4956 

5no 

5263 

54i7 

557i 

5725 

56 

3 

4ooo 

4176 

435i 

452? 

4702 

4878 

56 

4 

58?9 

6o33 

6187 

634i 

6496 

665o 

55 

4 

5o53 

5229 

54o5 

558i 

5757 

5933 

55 

5 

68o4 

6958 

7n3 

7267 

7422 

7576 

54 

5 

6109 

6285 

646i 

6638 

68i4 

6990 

54 

6 

773i 

7886 

8o4i 

8195 

835o 

85o5 

53 

6 

7167 

7343 

7520 

7697 

7874 

8o5o 

53 

7 

8660 

88i5 

8970 

9125 

928o 

9436 

52 

7 

8227 

84o4 

858i 

8759 

8936 

9u3 

52 

8 

959i 

9746 

9902 

..57 

.212 

.368 

5i 

8 

9290 

9468 

9645 

9823 

.... 

.178 

5i 

9 

io.86o524 

o679 

o835 

0991 

n46 

I3O2 

5o 

9 

10-920356 

o534 

0712 

o89o 

1068 

1246 

5o 

10 

i458 

1614 

1770 

I926 

2082 

2239 

49 

10 

1424 

1602 

1781 

J959 

2i38 

23i6 

49 

ii 

2395 

255i 

2708 

2864 

3O2O 

3i77 

48 

ii 

2495 

2673 

2852 

3o3i 

3210 

3389 

48 

12 

3333 

349o 

3647 

38o3 

396o 

4n7 

47 

12 

3568 

3747 

3926 

4io5 

4285 

4464 

47 

i3 

4274 

443  1 

4588 

4745 

4902 

5o59 

46 

i3 

4644 

4823 

5oo3 

5i83 

5362 

5542 

46 

i4 

52i6 

5374 

553i 

5688 

5846 

6oo3 

45 

i4 

5722 

5902 

6082 

6262 

644a 

6623 

45 

i5 

6161 

63i9 

6476 

6634 

6792 

695o 

44 

i5 

68o3 

6984 

7i64 

7345 

7525 

7706 

44 

16 

7107 

7265 

7423 

758i 

7739 

7898 

43 

16 

7887 

8068 

8249 

843o 

8611 

8792 

43 

J7 

8o56 

8214 

8372 

853i 

8689 

8848 

42 

*7 

8973 

9i54 

9336 

95i7 

9699 

9880 

42 

18 

9006 

9i65 

9324 

9482 

964i 

98oo 

4i 

18 

10.930062 

0244 

o425 

o6o7 

0789 

0971 

4i 

T9 

*9959 

.118 

.277 

.436 

.595 

•  754 

4o 

I9 

ix54 

i336 

i5i8 

1700 

i883 

2o65 

4o 

20 

10.870913 

1072 

1232 

i39i 

i55i 

I7IO 

39 

20 

2248 

243o 

26i3 

2796 

2979 

3162 

39 

21 

1870 

2029 

2i89 

2349 

25o8 

2668 

38 

21 

3345 

3528 

37n 

3894 

4o78 

4261 

38 

22 

2828 

2988 

3i48 

33o8 

3468 

3629 

37 

22 

4444 

4628 

48l2 

4995 

5i79 

5363 

37 

23 

3789 

3949 

4io9 

4270 

443o 

459i 

36 

23 

5547 

573i 

59i5 

6o99 

6283 

6467 

36 

24 

475i 

4912 

5o73 

5234 

5394 

5555 

35 

24 

6652 

6836 

7021 

72o5 

739o 

7575 

35 

25 

5716 

5877 

6o38 

6199 

636o 

6522 

34 

25 

7760 

7945 

8i3o 

83i5 

85oo 

8685 

34 

26 

6683 

6844 

7006 

7167 

7329 

749o 

33 

26 

8870 

9o56 

924l 

9427 

9612 

9798 

33 

27 

7652 

7813 

7975 

8i37 

8299 

846i 

32 

27 

9984 

.169 

.355 

.54i 

.727 

.9i4 

32 

28 

8623 

8785 

8947 

9109 

927i 

9433 

3i 

28 

10.941100 

1286 

1472 

i659 

i845 

2032 

3i 

29 

9596 

9758 

9921 

..83 

.246 

.4o8 

3o 

29 

22I9 

2406 

2592 

2779 

2966 

3i53 

3o 

3o 

10.880571 

0734 

0896 

loSg 

1222 

i385 

29 

3o 

334i 

3528 

37i5 

39O2 

4o9o 

4277 

29 

3i 

1  548 

1711 

1874 

2o38 

22OI 

2364 

28 

3i 

4465 

4653 

484i 

5o28 

52i6 

54o4 

28 

32 

2528 

2691 

2855 

3oi8 

3i82 

3345 

27 

32 

5593 

5781 

5969 

6157 

6346 

6534 

27 

33 

35o9 

3673 

3837 

4ooi 

4i65 

4329 

26 

33 

6723 

69I2 

7100 

7280 

7478 

7667 

26 

34 

4493 

4657 

4821 

4985 

5i5o 

53i4 

25 

34 

7856 

8o45 

8234 

8424 

86i3 

88o3 

25 

35 

5479 

5643 

58o8 

5972 

6i37 

63o2 

24 

35 

8992 

9l82 

937i 

956i 

975i 

994i 

24 

36 

6467 

6632 

6796 

6961 

7127 

7292 

23 

36 

io.95oi3i 

0321 

o5n 

0702 

0892 

io83 

23 

3? 

7457 

7622 

7787 

7953 

8118 

8284 

22 

37 

1273 

i464 

1  654 

i845 

2036 

2227 

22 

38 

8449 

86i5 

8781 

8946 

9112 

9278 

21 

38 

2418 

25o9 

2800 

2991 

3i83 

3374 

21 

39 

9444 

9610 

9776 

9942 

.108 

.274 

20 

39 

3566 

3757 

3949 

4i4i 

4332 

4524 

20 

4o 

10.890441 

0607 

o773 

o94o 

1106 

I273 

I9 

4o 

4716 

49o8 

5ioi 

5293 

5485 

5678 

I9 

4i 

i44o 

1606 

i773 

i94o 

2IO7 

2274 

18 

4i 

587o 

6o63 

6255 

6448 

664i 

6834 

18 

42 

244  1 

2608 

2775 

2942 

3no 

3277 

I? 

42 

7027 

7220 

74i3 

7606 

78oo 

7993 

J7 

43 

3444 

36i2 

3779 

3947 

4n5 

4282 

16 

43 

8187 

838o 

8574 

8768 

8961 

9i55 

16 

44 

445o 

46i8 

4786 

4954 

5l22 

529o 

i5 

44 

9349 

9544 

9738 

9932 

.126 

.321 

i5 

45 

5458 

5626 

5795 

5963 

6i3i 

63oo 

i4 

45 

10.960515 

0710 

ogoS 

1099 

I294 

1489 

i4 

46 

6468 

6637 

6806 

6974 

7i43 

73l2 

i3 

46 

i684 

1879 

2074 

2270 

2465 

2661 

i3 

4? 

748i 

765o 

7819 

7988 

8i57 

8326 

12 

47 

2856 

3o52 

3247 

3443 

3639 

3835 

12 

48 

8496 

8665 

8834 

9oo4 

9i73 

9343 

II 

48 

4o3i 

4227 

4424 

4620 

48i6 

5oi3 

II 

49 

95i3 

9683 

9852 

.  .22 

.192 

.362 

10 

49 

5209 

54o6 

56o3 

58oo 

5997 

6194 

10 

5o 

10.900532 

0702 

0873 

io43 

I2l3 

1  384 

9 

5o 

639i 

6588 

6785 

6983 

•7180 

7377 

9 

5i 

1  554 

1724 

i895 

2O66 

2236 

2407 

8 

5i 

7575 

7773 

797i 

8169 

8367 

8565 

8 

52 

2578 

2749 

292O 

3o9i 

3262 

3433 

7 

52 

8763 

896i 

9l59 

9358 

9556 

9755 

7 

53 

36o5 

3776 

3947 

4n9 

4290 

4462 

6 

53 

9954 

.162 

.35i 

.55o 

.749 

•  948 

6 

54 

4633 

48o5 

4977 

5i49 

532O 

5492 

5 

54 

io.97n48 

1  347 

1  546 

i?46 

1945 

2i45 

5 

55 

5664 

5837 

6oo9 

6181 

6353 

6526 

4 

55 

2345 

2545 

2745 

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60" 

50" 

40" 

30" 

20" 

10" 

a 

60"     50"   40" 

30" 

20"  I  10" 

(H 

Co-tangent  of  7  Degrees. 

§ 

Co-tangent  of  6  Degrees. 

,  <  1"  2"  3"  4"  5"  6"  7"  8"  9" 

P  Part  5  l"  2"  3"  4"  5"  6"   7"   8"   9" 

irt£  16  33  49  65  81  98  114  130  146 

m\  19  37  53  75  94  112  131  150  168 

.08 


LOGARITHMIC    b  i  N  E  s. 


.5 

Sine  of  84 

Degrees. 

a 

Sine  of  85  Degrees. 

* 

0"    i  10" 

20" 

30"  |  40" 

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20"  |  30" 

40" 

50" 

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60" 

50"  | 

40" 

30"   20"   10" 

. 

60"     50"   40"   30"   20"  i  10"  |  ^ 

Co-sine  of  5  Degrees. 

1 

Co-sine  of  4  Degrees.     1  S 

P  "po|-f  /     "* 

4" 

5//  6,/  r,  8/,  9,/  j       (  r/  „//  3//  4"  5//  G"  7"  8//  g// 

1 

1   1   1   2   2  I!       \  0   0   0   1   1   1   1   1   1 

LOGARITHMIC    TANGENT? 


10 


.s 

Tangent  of  84  Degrees. 

_s- 

Tangent  of  85  Degrees. 

i 

0" 

10" 

20" 

30" 

40" 

50" 

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49o7 

2 

57 

9943 

.242 

.542 

.84i 

n4i  i44o 

2 

58 

5i48 

5389 

563o 

587i 

6112 

6354 

I 

58 

ii  .  161740 

2041 

234l 

2642 

2942  3M3 

I 

59 

6596 

6837 

7079 

7321 

7563 

7806 

O 

59 

3545 

3846 

4i48 

4449 

4752  5o54 

O 

60" 

50"   40" 

30"   20" 

10" 

. 

60"    |  50" 

40"   30" 

20"   10/7 

g 

Co-tangent  of  5  Degrees. 

2 

Co-tangent  of  4  Degrees. 

£ 

p  p   $  1"  2"  3"  4"  5"  (5"  7"  8"  9" 
I  22  44  66  88  110  132  154  177  199 

p  p   $  1"  2"  3"  4"  5"  6"  7"  S"  9" 
ir  }  27  54  81  108  135  162  188  215  242 

I  10 


LOGARITHMIC   SINES. 


I 

Sine  of  86  Degrees. 

.S 

Sine  of  87  Degrees. 

s 

0" 

10" 

|  20"   30"   40" 

50" 

2 

0" 

10" 

20" 

30" 

40" 

50" 

0 

i 

9.99894i 
895o 

8942 
895i 

89448945|8947 
89538954!8955 

8948 
8957 

69 

58 

o 
i 

9.9994o4 
94n 

94o6 
94i2 

9407 

94i3 

94i4 

9409 

94i5 

94io 

5o 

58 

2 

8958 

896o 

8961  8963  8964 

8966 

57 

2 

94i8 

94i9 

9420 

9421 

9422 

9423 

57 

3 

8967 

8969 

8970  8971 

8973 

8974 

56 

f. 

9424 

9425 

9426 

9427 

9428 

943o 

56 

4 

8976 

8977,8979  8980 

8982 

8983 

55 

L 

943i 

9432 

9433 

9434 

9435 

9436 

55 

5 

8984 

8986898-78989 

899o 

8992 

54 

t 

9437 

9438 

9439 

944o 

944  1 

9442 

54 

6 

8993 

8995 

8996  8997 

8999 

9ooo 

53 

6 

9443 

9445 

9446 

9447 

9448 

9449 

53 

7 

9OO2 

9oo3 

9005  9006 

9oo9 

52 

7 

945o 

945i 

9452 

9453 

9454 

9455 

52 

8 

9OIO 

9oi2  9oi3  9oi5 

9016 

9oi7 

5i 

8 

9456 

9457 

9458 

9459 

9460 

946i 

5i 

9 

9019 

9O2O  9O22  9O23 

9024 

9026 

5o 

9 

9463 

9464 

9465 

9466 

9467 

9468 

5o 

10 

9.999027 

9o29  9o3o  9o32 

9o33 

9o34 

49 

10 

9  999469 

947Q 

94?i 

9472 

9473 

9474 

49 

ii 

9036 

9o37  9o39  9o4o 

9041 

9o43 

48 

ii 

9475 

9476 

9477 

9478 

9479 

948o 

48 

12 

904^ 

9o46  9o47  9o48 

9o5o 

9o5i 

47 

12 

948i 

9482 

9483 

9484 

9485 

9486 

47 

i3 

9o52 

9o54  9o55  9o57 

9o58 

9o59 

46 

i3 

9487 

9488 

9489 

949o 

949i 

9492 

46 

i4 

9061 

9o62;9o64  9o65 

9066 

9o68 

45 

i^ 

9493 

9495 

9496 

9497 

9498 

9499 

45 

i5 

9069 

907190729078 

9o75 

9o76 

44 

i5 

95oi 

9502 

95o3 

95o4 

95o5 

44 

16 

'  9°77 

9o79  9o8o  9o82 

9083 

9o84 

43 

16 

95o6 

95°7 

95o8 

95o9 

95io 

95n 

43 

17 

9086 

9087  9088  9o9o 

9091 

9o92 

42 

17 

95l2 

95i3 

95x4 

95i5 

95i6 

95i7 

42 

18 

9094 

9096  9097  9098 

9099 

9IOI 

4i 

18 

95i8 

95i9 

9520 

952I 

9522 

9523 

4i 

i9 

9102 

9io3  9105  9106 

9107 

9I09 

4o 

19 

9524 

9525 

9526 

9527 

9527 

9528 

4o 

20 

9.999110 

9111 

9113  9114 

9ii5 

9117 

39 

20 

9.999529 

953o 

953i 

9532 

9533 

9534 

39 

21 

9118 

9120  9121  9122 

9124 

9I25 

38 

21 

9535 

9536 

9537 

9538 

9539  954o 

38 

22 

9126 

9128  9129  9130 

9l32 

9i33 

37 

22 

954i 

9542 

9543 

9544 

9545 

9546 

37 

23 

9134 

9i36;9i379i38 

9i4o 

36 

23 

9547 

9548 

9549 

955o 

955i 

9552 

36 

24 

9142 

9i439i45 

9i46 

9i47 

9i49 

35 

24 

9553 

9554 

9555 

9556 

9557 

9557 

35 

25 

9i5o 

9i5i 

9i53 

9i54 

9i55 

9i57 

34 

25 

9558 

9559 

956o 

956i 

9562 

9563 

34 

26 
27 

9i58 
9166 

9l59 
9i67 

9161  9162 
9168  9170 

9i63 
9171 

9i65 

9I72 

33 

32 

26 
27 

9564 
957o 

9565 
957i 

9566 
9572 

9567 
9573 

9568 
9573 

9569 

9574 

33 

32 

28 

9174 

9i759i769i78 

9J79 

9i8o 

3i 

28 

9575 

9576 

9577 

9578 

9579 

958o 

3i 

29    9i8i 

9i839i849i85 

9187 

9i88 

3o 

29 

958i 

9582 

9583 

9584 

9585 

9586 

3o 

3o  ;  999  i  89 

9i9o 

9192  9193 

9i96 

29 

3o 

9.599586 

9587 

9588 

9589 

959° 

959i 

29 

3i 

9197 

9198  9199  92oi 

9202 

92o3 

28 

3i 

9592 

9593 

9594 

9595 

9596 

9597 

28 

32 

92o5 

9206^2079208 

92IO 

92II 

27 

32 

9597 

9598 

9599 

96oo 

96oi 

96o2 

27 

33 

92I2 

9213 

92i5|92i6 

9217 

92I9 

26 

33 

96o3 

96o4 

96o5 

96o6 

96o6 

96o7 

26 

34 

922O 

9221 

9222 

9224 

9225 

9226 

25 

34 

96o8 

96o9 

96io 

96n 

96l2 

96i3 

25 

35 

9227 

9229  9230 

9231 

9232 

9234 

24 

35 

96i4 

96i4  96i5 

96i6 

96i7 

96i8 

24. 

36 

9235 

92369237 

9239 

924o 

924l 

23 

36 

96i9 

962O  962I 

0,622 

9622 

9623 

23 

37 

9244  9245  9246 

9247 

9249 

22 

37 

9625 

9626 

9627 

9628 

9629 

22 

38 

9250 

925l 

9252  9254 

9255 

9256 

21 

38 

9629 

963o 

963i 

9632 

9633 

9634 

21 

39 

9257 

9258  926o|926i 

9262 

9263 

2O 

39 

9635 

9635 

9636 

9637 

9638 

9639 

2O 

4o 

9.990.265 

9266 

9267  9268 

9270927I 

19 

4o 

9.999640 

964i 

964i 

9642 

9643 

9644 

19 

4i 

9272 

9273  92749276 

92779278 

18 

4i 

9645 

9646 

9647 

9647 

9648 

9649 

18 

42 

9279 

9280  9282 

9283 

0.2849285 

ll 

42 

965o 

965i 

9652 

9653 

9653 

9654 

J7 

43 

9287 

92889289 

9290 

929I 

9293 

16 

43 

9655 

9656 

9657 

9658 

9658 

9659 

16 

44 

9294 

9295,9296 

9297 

9299 

93oo 

i5 

44 

9660 

966i 

9662 

9663 

9663 

9664 

i5 

45 

93oi 

93o2'93o3 

93o5 

93o6  93o7 

i4 

45 

9665 

9666 

9667 

9668 

9668 

9669 

i4 

46 

93o8 

9309 

9312 

93i3 

93i4 

i3 

46 

967o 

967i 

9672 

9672 

9673 

9674 

i3 

47 

93i5 

93i6!o:3i8 

93i9 

9320932I 

12 

47 

9675 

0676 

9677 

9677 

9678 

9679 

12 

48 

9322 

9323'9325 

9326 

93279328 

II 

48 

9680 

968i 

968i 

9682 

9683 

9684 

II 

49 

9329 

933i 

9332 

9333 

93349335 

10 

49 

9685 

9685 

9686 

9687 

9688 

9689 

IO 

5o 

9  999336 

93389339 

934o 

934i 

9342 

9 

5o 

9.999689 

969o 

969i 

9692 

9693 

9693 

9 

5i 

9343 

9344  9346 

9347 

9348  9349 

8 

5i 

9694 

9695 

9696 

9697 

9697 

9698 

8 

52 

935o 

935i 

9353 

9354 

93559356 

7 

52 

9699 

0.700 

97oo 

9701 

9702 

97o3 

7 

53 

9357 

9358 

9359 

936i 

9362  9363 

6 

53 

97o4 

97°5 

97o6 

97°7 

97°7 

6 

54 

9364 

9365 

9366 

9367 

9369937o 

5 

54 

9708 

9709 

97io 

97n 

9711 

97I2 

5 

55 

937i 

9372 

9373 

9374 

93759377 

4 

55 

97i3 

97i4 

97i4 

97i5 

97i6 

9717 

4 

56 

9378 

9379 

938o 

938i 

93829383 

3 

56 

9717 

97i8 

9719 

9720 

9720 

9721 

3 

57 

9384 

9385 

9387 

9388 

9389  939o 

2 

57 

9722 

9723 

9723 

9724 

9725 

9726 

2 

58 

939i 

9392 

9393 

9394 

93969397 

I 

58 

9726 

9727 

9728 

9729 

9729 

973o 

I 

59 

9398 

9399 

94oo  94oi 

94o2  94o3 

O 

59 

973i 

9732 

9732 

9733 

9734 

9735 

O 

60" 

50" 

40"  |  30"   20"  |  10" 

. 

CO" 

50"   40" 

30" 

20" 

10" 

j 

Co-sine  of  3  Degrees. 

§ 

Co-sine  of  2  Degrees. 

s 

(  1"  2"  3' 

'  4"  5"  6"  7"  8"  9" 

,  1"  2"  3"  4"  5"  6"  7"  &'  9" 

lrt{  0   0   0 

111111 

F.Fart^  0   0   0   0   0   1   1   1   1 

LOGARITHMIC    TANGENTS. 


11 


1 

Tangent  of  86  Degrees. 

| 

Tangent  of  87  Degrees. 

2 

0" 

10" 

20" 

30" 

40" 

50" 

m 

0" 

10" 

20" 

30" 

40" 

50" 

o 

n.  155356 

5659 

5962 

6265 

6568 

6872 

59 

o 

11.280604 

1007 

i4n 

1814 

22I9 

2623 

5  9 

I 

7175 

74?9 

7784 

8088 

8393 

8697 

58 

i 

3028 

3433 

3839 

4245 

4652 

5o58 

58 

2 

9002 

93o8 

96l3 

9919 

.224 

.53o 

57 

2 

5466 

5873 

6281 

6689 

7098 

75o7 

5? 

3 

11.160837 

n43 

i45o 

c 

I757 

2064 

237I 

56 

3 

7917 

8326 

8737 

9I47 

9558 

9970 

56 

4 

2679 

2987 

3295 

36o3 

3911 

4220 

55 

4 

11.290382 

0794 

1206 

i6i9 

2o33 

2446 

55 

5 

4529 

4838 

5i47 

5457 

5766 

6076 

54 

5 

2860 

3275 

369o 

4io5 

452i 

4937 

54 

6 

6387 

6697 

7008 

73i8 

7629 

7941 

53 

6 

5354 

577o 

6188 

66o5 

7024 

7442 

53 

7 

8252 

8564 

8876 

9188 

gSoo 

9813 

52 

7 

7861 

8280 

8700 

9120 

954i 

9962 

52 

8 

ii  .  170126 

o439 

0752 

1066 

i379 

1693 

5i 

8 

ii.3oo383 

o8o5 

1227 

1649 

2072 

2496 

5i 

9 

2008 

2322 

2637 

295l 

3267 

3582 

5o 

9 

2919 

3344 

3768 

4i93 

46i9 

5o44 

5o 

10 

3897 

42i3 

4529 

4845 

5i62 

5478 

A9 

IO 

547i 

5897 

6325 

6752 

7180 

76o8 

49 

ii 

5795 

6112 

643o 

6747 

7o65 

7383 

48 

1  1 

8037 

8466 

8896 

9326 

9756 

.187 

48 

12 

7702 

8020 

8339 

8658 

8977 

9T97 

47 

12 

i  i  .3/0619 

io5o 

i483 

1915 

2348 

2782 

47 

i3 

9616 

9936 

.256 

.577 

.897 

1218 

46 

i3 

32i6 

365o 

4o85 

452o 

4956 

5392 

46 

i4 

ii.  181539 

1860 

2182 

2D04 

2826 

3i48 

45 

i4 

5828 

6a65 

6702 

7140 

7578 

8017 

45 

i5 

347i 

3793 

4n6 

444o 

4763 

5087 

44 

i5 

8456 

8896 

9336 

9776 

.217 

.659 

44 

16 

54n 

5735 

6o59 

6384 

6709 

7034 

43 

16 

ii  .3  tnoo 

1  543 

i985 

2428 

2872 

33i6 

43 

*7 

7359 

7685 

Son 

8337 

8663 

8990 

42 

17 

376i 

4206 

465i 

5o97 

5543 

599o 

42 

18 

93i7 

9644 

9971 

.299 

.626 

.954 

4i 

18 

6437 

6885 

7333 

7782 

823i 

8680 

4i 

J9 

11.191283 

1611 

1940 

2269 

2598 

2928 

4o 

J9 

Qi3o 

958i 

..32 

.483 

.935 

1387 

4o 

20 

3258 

3588 

39i8 

4249 

4579 

4910 

39 

20 

u.3.Ji84o 

2293 

2747 

3aoi 

3656 

4m 

39 

21 

5242 

5573 

SgoS 

6237 

6569 

6902 

38 

21 

4567 

5o23 

548o 

5937 

6394 

6852 

38 

22 

7235 

7568 

7901 

8235 

8568 

8902 

37 

22 

73n 

7770 

8229 

8689 

9i5o 

96ii 

37 

23 

9237 

957i 

9906 

.241 

.577 

.912 

36 

23 

ii  .  ?4ooy2 

o534 

090,6 

i459 

I923 

2387 

36 

24 

II.  201248 

1  584 

1921 

2257 

2594 

2931 

35 

24 

285i 

33i6 

378i 

4247 

47i4 

5i8o 

35 

25 

3269 

36o6 

3944 

4282 

4621 

4960 

34 

25 

5648 

6116 

6584 

7o53 

7522 

7992 

34 

26 

5299 

5638 

5977 

63i7 

6657 

6997 

33 

26 

8463 

8933 

94o5 

9877 

.349 

.822 

33 

27 

7338 

7679 

8020 

836i 

87o3 

9o45 

32 

27 

u.S5i2o6 

1770 

2244 

27i9 

3i95 

367i 

32 

28 

9387 

9729 

..72 

.4i5 

.758 

1102 

3i 

28 

4i47 

4624 

5l02 

558o 

6o59 

6538 

3i 

29 

n.2ii446 

1790 

2134 

2479 

2823 

3i69 

3o 

29 

7018 

7498 

7979 

846o 

8942 

9424 

3o 

3o 

35i4 

386o 

4206 

4552 

4898 

5245 

29 

3o 

99°7 

.390 

•  874 

i359 

1  844 

2329 

29 

3i 

5592 

5939 

6287 

6635 

6983 

733i 

28 

3i 

ii  .362816 

33o2 

3789 

4277 

4765 

5254 

28 

32 

7680 

8o29 

8378 

8728 

9o78 

9428 

27 

32 

5744 

6234 

6724 

72i5 

77°7 

8i99 

27 

33 

9778 

.  I29 

.48o 

.83i 

n83 

i534 

26 

33 

8692 

9i85 

9679 

.i73 

.668 

n64 

26  ' 

34 

11.221886 

2239 

259I 

2944 

3298 

365i 

25 

34 

11.371660 

2i56 

2654 

3i5i 

365o 

4i4g 

25: 

35 

4oo5 

4359 

47i3 

5o68 

5423 

5778 

24 

35 

4648 

5i48 

5649 

6i5o 

6652 

7i54 

24 

36 

6i34 

6489 

6845 

7202 

7558 

79i5 

23 

36 

7657 

8161 

8665 

9170 

9675 

.18! 

23 

3? 

8273 

863o 

8988 

9346 

97o5 

..63 

22 

37 

11.380687 

1194 

I7O2 

22IO 

2719 

3228 

22 

38 

II  .23o422 

0782 

n4i 

i5oi 

1861 

2222 

21 

38 

3738 

4249 

476o 

5272 

5785 

6298 

21 

39 

2583 

2944 

33o5 

3667 

4029 

439I 

2O 

39 

.  6811 

7325 

784o 

8356 

8872 

9388 

2O 

4o 

4?54 

5n6 

548o 

5843 

62O7 

657i 

19 

4o 

99o6 

.424 

.942 

i46i 

1981 

25oi 

I9 

4i 

6935 

73oo 

7665 

8o3o 

8396 

8762 

18 

4i 

II  ,393O22 

3544 

4o66 

4589 

5n3 

5637 

iS 

42 

9128 

9495 

986i 

.229 

.596 

.964 

17 

42 

6161 

6687 

72l3 

774o 

8267 

8795 

*7 

43 

II.24l332 

1700 

2069 

2438 

28o7 

3i77 

16 

43 

9323 

9853 

.382 

.9i3 

i444 

!976 

16 

44 

3547 

39i7 

4288 

4659 

5o3o 

54oi 

i5 

44 

ii.4o25o8 

3o4i 

3575 

4no 

4645 

5i8o 

i.5 

45 

5773 

6i45 

65i8 

6891 

•7264 

7637 

i4 

45 

6717 

6254 

6792 

733o 

7869 

84o9 

i4 

46 

Son 

8385 

8759 

9i34 

95o9 

9884 

i3 

46 

894Q 

949o 

..32 

•  574 

1117 

1661 

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4? 

ii  .250260 

o636 

IOI2 

i389 

i766 

2143 

12 

47 

II  .4l22o5 

2751 

3296 

3843 

439o 

4938 

12 

48 

2521 

2899 

3277 

3656 

4o34 

44i4 

I  I 

48 

5486 

6o3fi 

6586 

7i36 

7688 

8240 

II 

49 

4793 

5i73 

5553 

5934 

63i5 

6696 

IO 

49 

6792 

9346 

9900 

vr: 

IOIO 

i566 

10 

5o 

7078 

746o 

7842 

8224 

86o7 

8991 

9 

5o 

ii  .422123 

2681 

324013799 

4358 

4919 

9 

5i 

9374 

9758 

.142 

.527 

.9I2 

1297 

8 

5i 

548o 

60^2 

660517168 

7733 

8298 

8 

52 

11.261683 

2069 

2455 

2842 

3229 

36i6 

7 

52 

8863 

943o 

9997 

.565 

n33 

I7O2 

7 

53 

4oo4 

4392 

4780 

5i69 

5558 

5947 

6 

53 

11.432273 

2843 

34i5 

3987 

456o 

5i34 

6 

54 

6337 

6727 

7117 

75o8 

7899 

8291 

5 

54 

57o9 

6284 

6860 

7437 

8oi5 

8593 

5 

55 

8683 

9075 

0.467 

986o 

.254 

.647 

4 

55 

9172 

9752 

.333 

.9i5 

1497 

2080 

4 

56 

11.271041 

i435 

i83o 

2225 

2620 

3oi6 

3 

56 

11.442664 

3248 

3834 

442O 

5007 

5595 

3 

57 

3412 

3809 

4206 

46o3 

5ooo 

5398 

2 

57 

6i83 

6773 

7363 

7954 

8546 

9i38 

2 

58 

5796 

6195 

6594 

6993 

7393 

7793 

I 

58 

9732 

.326 

.921 

1517 

2Il3 

2711 

I 

59 

8194 

8595 

8996 

9397 

9799 

.202 

O 

59 

ii.4533o9 

3908 

45o8 

5  1  09 

57n 

63i3 

O 

60" 

50"  |  40" 

30" 

20" 

10" 

c 

60"     50"   40"  |  30"  |  20"   10" 

f* 

Co-tangent  of  3  Degrees. 

§ 

Co-tangent  of  2  Degrees. 

S 

p  p  .  <  1"  2"  3"  4"  5"  6"  7"  8"  H" 
1  35  69  104  138  173  207  242  27fi  311 

p  p   C  1"  2"  3"  4"  5"  6"  7"  8"  9" 
\  48  97  14*5  193  242  290  338  387  435 

LOGARITHMIC    SINES 


A 

Sine  of  88  Degrees. 

4 

Sine  of  89  Degrees. 

m 

0" 

10" 

20" 

30" 

40" 

50" 

a 

0" 

10" 

20" 

30" 

40" 

50" 

o 

9.999735 

9736 

9737 

9738 

9738 

9739 

59 

0 

9.999934 

9934 

9935 

9935 

9935 

9936 

59 

i 

9?4o 

974o 

974i 

974s 

9743 

9743 

58 

I 

9936 

9936 

9937 

9937 

9937 

9938 

58 

2 

9744 

9745 

9746 

9746 

9747 

9748 

57 

2 

9938 

9939 

9939 

9939 

9940 

994o 

57 

3 

9748 

9749 

975o 

975i 

975i 

9752 

56 

3 

994o 

994i 

994i 

994i 

9942 

9942 

56 

4 

9753 

9753 

9754 

9755 

9756 

9756 

55 

4 

9942 

9943 

9943 

9943 

9944 

9944 

55 

5 

9757 

9758 

9758 

9759 

976o 

976o 

54 

5 

9944 

9945 

9945 

9945 

9946 

9946 

54 

6 

9761 

9762 

9763 

9763 

9764 

9765 

53 

6 

9946 

9947 

9947 

9947 

9948 

9948 

53 

7 

9765 

9766 

9767 

9767 

9768 

9769 

52 

7 

9948 

9949 

9949 

9949 

995o 

995o 

52 

8 

9769 

977° 

9771 

9772 

9772 

9773 

5i 

8 

995o 

995i 

995i 

995i 

9952 

9952 

5i 

9 

9774 

9774 

9775 

9776 

9776 

9777 

5o 

9 

9952 

9953 

9953 

9953 

9953 

9954 

5o 

JO 

9.999778 

9778 

9779 

9780 

978o 

978i 

49 

10 

9.999954 

9954 

9955 

9955 

9955 

9956 

49 

ii 

9782 

9782 

97839784 

9784 

9785 

48 

ii 

9956 

9956 

9956 

9957 

9957 

9957 

48  | 

12 

9786 

9786 

97879788 

9788 

9789 

47 

12 

9958 

9958 

9958 

9959 

9959 

9959 

4? 

i3 

979° 

979° 

979I|9792 

9792 

9793 

46 

i3 

9959 

996° 

996o 

996o 

996i 

996i 

46 

i4 

9794 

9794 

97959795 

9796 

9797 

45 

i4 

9961 

996i 

9962 

9962 

9962 

9963 

45 

i5 

9797 

9798 

9799  9799 

98oo 

98oi 

44 

i5 

9963 

9963 

9963 

9964 

9964 

9o64 

44 

16 

9801 

9802 

98o398o3 

98o4 

98o4 

43 

16 

9964 

9965 

9965 

9965 

9965 

9966 

43 

17 

98o5 

9806 

98o6  98o7 

98o8 

9808 

42 

J7 

9966 

9966 

9967 

9967 

9967 

9967 

42 

18 

9809 

9809 

98io 

98n 

98n 

98l2 

4i 

18 

9968 

9968 

9968 

9968 

9969 

9969 

4i 

I-9 

9813 

9813 

98i4 

98i4 

98i5 

98i6 

4o 

'9 

9969 

9969 

9970 

997° 

997° 

997° 

4o 

20 

9.999816 

9817 

9817 

98i8 

98i9 

9819 

39 

20 

9-999971 

997i 

997i 

9971 

9972 

9972 

39 

21 

9820 

9820 

9821 

9822 

9822 

9823 

38 

21 

9972 

9972 

9973 

9973 

9973 

9973 

38 

22 

9824 

9824 

9825 

9825 

9826 

9827 

37 

22 

9973 

9974 

9974 

9974 

9974 

9975 

37 

23 

9827 

9828 

9828 

9829 

9829 

983o 

36 

23 

9975 

9975 

9975 

9976 

9976 

9976 

36 

24 

983i 

9831 

98329832 

98339834 

35 

24 

9976 

9976 

9977 

9977 

9977 

9977 

35 

25 

9834 

9835 

98359836 

9836  9837 

34 

25 

9977 

9978 

9978 

997s 

9978 

9979 

34 

26 

9838 

9838 

98399839 

984o 

9840 

33 

26 

9979 

9979 

9979 

9979 

9980 

9980 

33 

27 

9841 

9842 

9842^843 

9843 

9844 

32 

27 

9980 

998o 

998o 

998i 

9981 

9981 

32 

28 

9844 

9845 

9846  9846 

9847 

9847 

3i 

28 

9981 

998i 

9982 

9982 

9982 

9982 

3i 

29 

9848 

9848 

9849  9849 

985o 

985i 

3o 

29 

9982 

9983 

9983 

9983 

9983 

9983 

3o 

3o 

9.099851 

9852 

9852  9853 

9853 

9854 

29 

3o 

9.999983 

9984 

9984 

9984 

9984 

9984 

29 

3i  '  "9854 

9855 

98569856 

9857 

9857 

28 

3i 

9985 

9985 

9985 

9985 

9985 

9985 

28 

32      9868 

9858 

98599859 

986o 

986o 

27 

32 

9986 

9986 

9986 

9986 

9986 

9986 

27 

33 

9861 

986i 

9862  9863 

9863 

9864 

26 

33 

9987 

9987 

9987 

9987 

9987 

9987 

26 

34 

9864 

9865 

9865  9866 

9866 

9867 

25 

34 

9988 

9988 

9988 

9988 

9988 

9988 

25 

35 

9867 

9868 

9868:9869 

9869 

987o 

24 

35 

9989 

9989 

9989 

9989 

9989 

9989 

24 

36 

9870 

987i 

987i  9872 

9872 

9873 

23 

36 

9989 

999° 

999° 

999° 

999° 

999° 

23 

37 

9873 

9874 

98749875 

9875 

9876 

22 

37 

9990 

999° 

9991 

9991 

9991 

9991 

22 

38 

9876 

9877 

98779878 

9878 

9879 

21 

38 

9991 

999  ' 

9991 

9992 

9992 

9992 

21 

39 

9879 

988o 

988o988i 

9881 

9.882 

2O 

39 

9992 

9992 

9992 

9992 

9992 

9993 

20 

4o 

9.999882 

9883 

9883  9884 

9884 

9885 

I9 

4o 

9-9.99993 

9993 

9993 

9993 

9993 

9993 

'9 

4i 

9885 

9886 

98869887 

9887 

9888 

18 

4i 

9993 

9993 

9994 

9994 

9994 

9994 

18 

4s 

9888 

9889 

9889  989o 

9890 

989i 

!7 

42 

9994 

9994 

9994 

9994 

9994 

9995 

17 

43 

9891 

9892 

9892  9892 

9893 

9893 

16 

43 

9995 

9995 

9995 

9995 

9995 

9995 

16 

44 

9894 

9894 

9895  98c5 

9896 

9896 

i5 

44 

9995 

9995 

9995 

9996 

9996 

9996 

i5 

45 

9897 

9897 

9898  9898 

9898 

9899 

i4 

45 

9996 

9996 

9996 

9996 

9996 

9996 

i4 

46 

9899 

9900 

99oo  99oi 

9901 

9902 

i3 

46 

9996 

9996 

9997 

9997 

9997 

9997 

i3 

47 

9902 

99°3 

99o3  99o3 

9904 

9904 

12 

47 

9997 

9997 

9997 

9997 

9997 

9997 

12 

48 

99o5 

99o5 

99o6  99o6 

9906 

99°7 

II 

48 

9997 

9997 

9997 

9998 

9998 

9998 

II 

49 

9907 

9908 

99o8  99o9 

9909 

9910 

10 

49 

9998 

9998 

9998 

9998 

9998 

9998 

10 

5o 

9.999910 

99io 

99ii99n 

9912 

9912 

9 

5o 

9.999998 

9998 

9998 

9998 

999s 

9998 

9 

5i 

99i3 

99i3 

99i3  99i4 

9914 

99i5 

8 

5i 

9999 

9999 

9999 

9999 

9999 

9999 

8 

52 

99i5 

99i5 

99i699i6 

99i7 

99i7 

7 

52 

9999 

9999 

9999 

9999 

9999 

9999 

7 

53 

99i8 

99i8 

99189919 

99I9 

992O 

6 

53 

9999 

9999 

9999 

9999 

9999 

9999 

6 

54 

9920 

992O 

9921 

992I 

9922 

9922 

5 

54 

9999 

9999 

9999 

9999 

9999 

.... 

5 

55 

9922 

9923 

9923 

9924 

9924 

9924 

4 

55 

o.oooooo 

oooo 

oooo 

oooo 

oooo 

oooo 

4 

56 

9925 

9925 

9926 

9926 

9926 

9927 

3 

56 

oooo 

oooo 

oooo 

oooo 

oooo 

oooo 

3 

5? 

9927 

9927 

9928 

9928 

9929 

9929 

2 

57 

oooo 

oooo 

oooo 

oooo 

oooo 

oooo 

2 

58 

9929 

993o 

993o 

993i 

993i 

993i 

I 

58 

oooo 

oooo 

oooo 

oooo 

oooo 

oooo 

I 

h 

9932 

9932 

9932 

9933 

9933 

9933 

O 

59 

oooo 

oooo 

oooo 

oooo 

oooo 

oooo 

O 

60" 

50" 

40" 

30" 

20" 

10" 

C 

60" 

50" 

40" 

30" 

20" 

10" 

a 

Co-sine  of  1  Degree. 

£ 

Co-sine  of  0  Degree. 

i 

A  I"  2"  3"  4"  5"  6"  7''  8"  9" 
irt{  000000001 

.(  1"  2"  3"  4"  5"  6''  7'  8"  9" 
""^0   0   00   0   0   000 

LOGARITHMIC    TANGENTS. 


1  1  * 


A 

Tangent  of  88  Decrees. 

P.  Part 

& 

0" 

10" 

20" 

30" 

40" 

50" 

tol". 

0 

11.456916 

11.457520 

ii.  458i25 

u.45873i 

11.459338 

11.459945 

59 

60.6 

I 

46o553 

46n63 

46i773 

462383 

462995 

4636o8 

58 

61.1 

2 

464221 

464836 

46545i 

466067 

466684 

467302 

57 

61.6 

3 

467920 

46854o 

469i6o 

469782 

470404 

471027 

56 

62.2 

4 

47i65i 

472276 

4729O2 

473528 

474:56 

474785 

55 

62.7 

5 

47$4i4 

476044 

476676 

477308 

477941 

478575 

54 

63.3 

6 

479210 

479846 

480482 

481120 

48i759 

482398 

53 

63,8 

7 

483o39 

48368o 

484323 

484966 

4856n 

486256 

52 

64.4 

8 

486902 

487549 

488i98 

488847 

489497 

490148 

5i 

65.o 

9 

490800 

49i453 

492io7 

492762 

4934i8 

494075 

5o 

65.5 

10 

H.494733 

n  .495392 

ii  .496o52 

i  i  .496713 

11.497375 

11,498038 

49 

66.1 

1  1 

498702 

499367 

5ooo33 

500700 

5oi368 

502037 

48 

66.8 

12 

502707 

5o3378 

5o4o5i 

504724 

SoSSgS 

506073 

47 

67.4 

i3 

506750 

507427 

5o8io6 

508785 

5o9466 

5ioi48 

46 

68.0 

i4 

5io83o 

5n5i4 

5i2i99 

5i2885 

5i3572 

5i426o 

45 

68.6 

i5 

5i495o 

*5i564o 

5i633i 

517024 

517717 

5i84i2 

44 

69.3 

16 

519108 

SigSoS 

52o5o3 

52I2O2 

521903 

522604 

43 

70.0 

*7 

523307 

524oio 

524}i5 

525421 

526128 

526837 

42 

70.7 

18 

527546 

528257 

528969 

529682 

53o396 

53iin 

4i 

7i.3 

19 

53i828 

532545 

533264 

533984 

5347o5 

535428 

4o 

72.7 

20 

n.536i5i 

ii.  536876 

11.537602 

ii.53833o 

ii.  SSgoSS 

11.539788 

39 

72.8 

21 

54o5i9 

54i25i 

54i984 

542719 

543455 

544192 

38 

73.5 

22 

544g3o 

545670 

5464i  i 

547i53 

547896 

54864i 

37 

74.3 

23 

549387 

55oi34 

55o883 

55i632 

55a384 

553i36 

36 

75.0 

24 

55389o 

554645 

5554oi 

556i59 

556918 

557678 

35 

75.8 

25 

55844o 

559203 

559967 

56o733 

56i5oo 

562268 

34 

76.6 

26 

563o38 

563809 

56458i 

565355 

566i3o 

566907 

33 

77.5 

27 

567685 

568464 

569245 

570027 

570811 

571596 

32 

78.3 

28 

572382 

573170 

573959 

574750 

575542 

576336 

3i 

79.i 

29 

577i3i 

5779a8 

578726 

579525 

58o326 

58ii28 

3o 

80  o 

3o 

11.581932 

11.582737 

n.583544 

n.584353 

n.585i63 

n.585974 

29 

80.  9 

3i 

586787 

587601 

5884i7 

589235 

590054 

590874 

28 

81.8 

32 

591696 

592520 

593345 

594172 

595000 

59583o 

27 

82.8 

33 

596662 

597495 

59833o 

599166 

6oooo4 

6oo844 

26 

83.  7 

34 

6oi685 

602528 

6o3372 

604218 

6o5o66 

605915 

25 

84-7 

35 

606766 

607619 

608474 

609330 

610187 

611047 

24 

85.  7 

36 

611908 

612771 

6i3636 

6i45o2 

615370 

616240 

23 

86.  7 

37 

617111 

617985 

618860 

6i9737 

620615 

621496 

22 

87.8 

38 

622378 

623262 

624147 

625o35 

625924 

626810 

21 

88.8 

39 

627708 

628603 

6?.95oo 

630399 

631299 

632201 

20 

89-9 

4o 

n.633io5 

ii.  634oi2 

n.6349i9 

11.635829 

11.636741 

n.637655 

'9 

91.1 

4i 

63857o 

639488 

640407 

641329 

642252 

643i77 

iS 

92.2 

42 

644io5 

645o34 

645965 

646899 

647834 

648771 

i? 

93.4 

43 

649711 

65o65a 

65i595 

65254i 

653488 

654438 

16 

94.6 

44 

655390 

656343 

657299 

658257 

659217 

660179 

i5 

95.9 

45 

66n44 

662110 

663o79 

664o5o 

665023 

665998 

i4 

97.2 

46 

666975 

667'j55 

668936 

669920 

6709O7 

671895 

i3 

98.5 

47 

672886 

673S79 

674874 

675871 

676871 

677873 

12 

99.8 

48 

678878 

679885 

68o894 

681905 

6829i9 

683935 

II 

loi.S 

49 

684954 

685975 

686998 

688024 

689o52 

690083 

IO 

102.7 

5o 

ii  .691116 

n.692i5i 

ii  .693i89 

ii  .694230 

ii.695273 

11.696318 

9 

104.2 

5i 

697366 

6984i7 

69947o 

700526 

7oi584 

702645 

8 

io5.7 

5- 

703708 

704774 

705843 

706914 

707988    7O9o65 

7 

107.2 

53 

710144 

711226 

7i23n 

713398 

7i4488 

7i558i 

6 

io8.9 

54 

716677 

717775 

718876 

719980 

721087 

722196 

5 

no.  5 

55 

723309 

724424 

725542 

726663 

727787 

728914 

4 

112  2 

56 

73oo44 

731176 

7323i2 

73345i 

734592 

735737 

3 

n4.o 

57 

736885 

738o35 

739189 

74o346 

74i5o6 

742669 

2 

u5.8 

58 

743835 

745oo4 

746177 

747352 

74853i 

7497i3 

I 

117.7 

59 

750898 

752087 

753279 

754474 

755672 

756874 

0 

n9.7 

eo" 

50" 

40" 

30"         20"         10" 

pj 

Co-tangent  of  1  Degree. 

g 

114 


AUXILIARY    TABLE    FOR    SINES,    &  c. 


i 

DD 

i 

0  Degree. 

i 

J        1  Degree. 

g 

0 

.3 

log.  sin.  A  — 

log.  tan.  A— 

log.  cot  A-f 

0 

c 

01;.  sin.  A— 

log.  tan.  A— 

log.  cot  A-f 

TO 

log.  A''. 

log.  A". 

log.  A". 

• 

9 

log.  A". 

log.  A". 

log.  A". 

O 

0 

4.685575 

4.685575 

i5.3i4425 

60 

36oo 

0 

4.685553 

4.686619 

i5.3i438i 

60 

60 

i 

575 

575 

426 

59 

366o 

I 

552 

620 

38o 

59 

I2O 

2 

575 

575 

426 

58 

3720 

2 

55i 

622 

378 

58 

1  80 

Q 

575 

575 

426 

57 

378o 

3 

55i 

623 

377 

57 

24o 

4 

575 

575 

426 

56 

384o 

4 

55o 

626 

375 

56 

3oo 

c 

575 

575 

426 

55 

39oo 

5 

549 

627 

373 

55 

36o 

6 

575 

575 

426 

54 

396o 

6 

548 

628 

372 

54 

420 

7 

575 

575 

4a5 

53 

4O2O 

7 

547 

63oj      370 

53 

48o 

8 

574 

576 

424 

52 

4o8o 

8 

54? 

632I      368 

52 

54o 

9 

574 

576 

424 

5i 

4i4o 

9 

546 

6331     36? 

5i 

600 

10 

4.685574 

4.685576 

i5.3i4424 

5o 

4200 

10 

4.685545 

4.685635 

i5.3i4365 

5o 

660 

ii 

574 

576 

424 

49 

4260 

ii 

544 

637 

363 

49 

72O 

12 

574 

577 

423 

48 

4320 

12 

543 

638 

362 

48 

780 

i3 

574 

577 

423 

47 

438o 

1  3'     542 

64o 

36o 

47 

84o 

i4 

574 

577 

423 

46 

444o 

i4 

54i 

.642 

358 

46 

900 

i5 

573 

578 

422 

45 

45oo 

i5 

54o 

644 

356 

45 

960 

16 

573 

578 

422 

44 

456o 

<6 

539 

646 

354 

44 

1  020 

17 

573 

578 

422 

43 

4620 

17 

539 

647 

353 

43 

1080 

18 

573 

5  79 

421 

42 

468o 

18 

538 

649 

35i 

42 

n4o 

19 

573 

579 

421 

4i 

474o 

19 

537 

661 

349 

4i 

1  200 

20 

4.685572 

4.68558o 

i5.3i442o 

4o 

4800 

20 

4.685536 

4.685653 

i5.3i4347 

4o 

1260 

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og.  cos.  A  — 

og.  eot.  A— 

log.  tan.  A-f 

i 

log.  COB.  A  — 

log.  cot  A  — 

log.  tan.  A-f 

• 

log.  c.  A". 

log.  c.  A". 

log.  c.  A." 

* 

log.  c.  A". 

log.  c.  A". 

log.  c.  A". 

1 

89  Degrees. 

% 

88  Degrees. 

'i 

LOGARITHMIC    TANGENTS, 


1  15 


1 

Tangent  of  89  Degrees.          ,      | 

P.  Part 

i  s 

0" 

10" 

20" 

30" 

40" 

50" 

to  1". 

o 

ii  .758079 

ii  .759287 

11.760498 

1  1  .761714 

II  .762932 

II  .764l54 

69 

121  .7 

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765379 

766608 

767840 

76(^076 

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77i558 

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777826 

779091 

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3    78o359 

781631 

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784186 

785470 

786757 

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4    788047 

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790641 

79i943 

79325o 

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53 

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7 

811964 

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816081 

817462 

818847 

52 

137-9 

8 

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824434 

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9 

828672 

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834387 

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143.4 

10 

11.837273 

11.838724 

11.840179 

ii.  841639 

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11.844674 

49 

146.2 

ii 

846o48 

847528 

849013 

85o5o3 

85i999 

853499 

48 

149.3 

12 

855oo4 

8565i5 

858o3i 

859553 

861079 

862611 

47 

162.4 

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8656^2 

867240 

868794 

870354 

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873490 

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876649 

878237 

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45 

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886266 

887890 

8895i9 

89n55 

44 

162.7 

16 

892797 

894446 

896100 

897761 

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43 

166.4 

17 

902783 

904470 

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907863 

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170.3 

18 

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916464 

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174.4 

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178.7 

20 

11.934194 

11.936008 

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11.939658 

11.941494 

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39 

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21 

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33 

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222.7 

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29 

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237.3 

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12  .o6i56i 

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12.066441 

12.068902 

12.071377 

29 

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28 

264.0 

32 

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102228 

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25 

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296.4 

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i65i99 

168290 

171404 

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37 

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180880 

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22 

321.7 

38 

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197148 

200476 

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207210 

210616 

21 

336.7 

39 

214049 

217510 

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228060 

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353.1 

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12.235239 

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12.242538 

12.246235 

12  249963 

12.253723 

19 

371.2 

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18 

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42 

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17 

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318787 

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327660 

16 

438.  7 

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629.4 

49 

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12.543572 

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12  566236 

12.674061 

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14.013395 

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60" 

50" 

40" 

30" 

20" 

10" 

. 

Co-tangent  of  0  Degree. 

* 

116 


NATURAL    SINES. 


£ 

X 

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4° 

5° 

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Natural  Co-sines. 

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4.78 

NATURAL    TANGENTS. 


1  17 


j 

£ 

0° 

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2° 

3° 

4° 

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7° 

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87° 

86° 

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83°   82° 

81° 

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pi 

Natural  Co-tangents. 

2 

£l*>85 

4.85 

4.86 

4.87 

4.88 

4.89 

4.91   4.93 

4.96 

4.98 

lib 


NATURAL    SINES. 


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1 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

—  -"~-  1 

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6 

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79° 

78° 

77° 

76° 

75° 

74° 

73° 

72°   71° 

70°   . 

Natural  Co-sines. 

fo^-77 

4.75 

4.73 

4*71 

4.69 

4.67 

4.65 

4.62 

4.6o 

4.57 

i 

NATURAL    TANGENTS. 


I 

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j° 

11° 

12° 

13° 

14° 

15° 

16° 

17°  |  18° 

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56 

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6 

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9821 

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6281 

54 

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6494 

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53 

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52 

9 

9028 

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3627 

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9697 

8820 

8i57 

7724 

I9 

42 

8952 

7090 

536o 

3774 

2345 

108-7 

3oooi4 

9141 

848  1 

8o52 

18 

43 

9253 

7393 

5665 

4082 

2666 

i4oi 

o33i 

9461 

8806 

838o 

r7 

44 

9555 

7697 

597J 

4390 

296-7 

I7i5 

0649 

9782 

9i3o 

8708 

16 

45 

9856 

8000 

6277 

4698 

3278 

2029 

0966 

320io3 

9454 

9o37 

i5 

46 

190157 

83o4 

6583 

5007 

3589 

2343 

1283 

0423 

9779 

9365 

i4 

47 

0459 

8607 

6889 

53i5 

3900 

2657 

1600 

o744 

34oio3 

9694 

i3 

48 

0760 

8911 

7194 

5624 

4211 

297i 

1918 

io65 

0428 

360022 

12 

49 

1062 

9214 

75oo 

5932 

4523 

3286 

2235 

i386 

O752 

o35i 

II 

5o 

I9i363 

209518 

2278o6 

246241 

264834 

2836oo 

302553 

32I707 

34io77 

360679 

IO 

5i 

i665 

9822 

8112 

6549 

5i45 

39i4 

287O 

2028 

1402 

1008 

9 

62 

1966 

210126 

84i8 

6858 

5457 

4229 

3i88 

2349 

I727 

i337 

8 

53 

2268 

0429 

8724 

7166 

5768 

4543 

35o6 

2670 

2O52 

1666 

7 

54 

2570 

o733 

9o3i 

7475 

6o79 

4857 

3823 

2991 

2377 

1995 

6 

55 

2871 

1037 

9337 

7784 

6391 

5l72 

4i4i 

33i2 

2702 

2324 

5 

56 

3i73 

i34i 

9643 

8092 

6702 

5487 

4459 

3634 

3027 

2653 

4 

57 

3475 

1  645 

9949 

8401 

7oi4 

58oi 

4777 

SgSS 

3352 

2982 

3 

58 

3777 

1949 

230255 

87io 

7326 

6116 

5o95 

4277 

3677 

33i2 

2 

59 

4o78 

2253 

o562 

9019 

7637 

643  1 

54i3 

4598 

4OO2 

364i 

I 

79° 

78°   77° 

76° 

75° 

74° 

73° 

72°   71° 

70° 

a* 

Natural  Co-tangents. 

§ 

p  P 

££*•«' 

5.o5   5.c9 

5.i3 

5.i7 

5.22 

5.27 

5.33 

5.39 

5.46 

120 


NATURAL    SINES. 


.s 
1 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

0 

342020 

358368 

374607 

39o73i 

4o6737 

422618 

43837i 

45399o 

469472 

484810 

60 

I 

•2293 

864o 

4876 

°999 

7OO2 

2882 

8633 

425o 

9728 

5o64 

59 

2 

2567 

89n 

5i46 

1267 

7268 

3i45 

8894 

45o9 

9985 

53i8 

58 

3 

2840 

9i83 

54i6 

i534 

7534 

34o9 

9i55 

4768 

470242 

5573 

57 

4 

3n3 

9454 

5685 

1802 

7799 

3673 

94i7 

6027 

0499 

5827 

56 

5 

3387 

9725 

5955 

2070 

8o65 

3936l   9678 

5286 

0755 

6081 

55  i 

6 

366o 

9997 

6224 

a337 

833o 

4199 

9939 

5545 

IOI2 

6335 

54 

7 

3933 

360268 

6494 

26o5 

8596 

4463 

44O2OO 

58o4 

1268 

6590 

53 

8 

4206 

o54o 

6763 

2872 

8861 

4726 

0462 

6o63 

i525 

6844 

52 

9 

4479 

0811 

7o33 

3i4o 

9I27 

4990 

0723 

6322 

1782 

7o98 

5i 

10 

344752 

361082 

3773o2 

3934o7 

4o9392 

4252531  44o984 

45658o 

472038 

487352 

5o 

ii 

5o25 

i353 

757i 

3675 

9658 

55i6[   1245 

6839 

2294 

•7606 

49 

12 

5298 

i625 

784i 

3942 

9923 

5779|   i5o6 

7o98 

255i 

786o 

48 

i3 

557i 

1896 

8110 

42O9 

4ioi88 

6042 

1767 

7357 

2807 

8n4 

47 

i4 

5844 

2l67 

8379 

4477 

o454 

63o6 

2028 

76i5 

3o63 

8367 

46 

i5 

6n7 

2438 

8649 

4744 

°7I9 

6569 

2289 

7874 

3320 

8621 

45 

16 

639o 

2709 

8918 

Son 

0984 

6832 

255o 

8i33 

3576 

8875 

44 

17 

6663 

298o 

9187 

5278 

1249 

7o95 

2810 

839i 

3832 

9I29 

43 

18 

6936 

325i 

9456 

5546 

i5i4 

7358 

3071 

865o 

4o88 

9382 

4a 

i9 

7208 

3522 

9725 

58i3 

1779 

•7621 

3332 

89o8 

4344 

9636 

4i 

20 

34748i 

363793 

379994 

396o8o 

4i2o45 

427884 

443593 

459i66 

474600 

48989o 

4o 

21 

7754 

4o64 

380263 

6347 

a3io 

8i47 

3853 

9425 

4856 

49oi43 

39 

22 

802-7 

4335 

o532 

6614 

2575 

84io 

4n4 

9683 

5lI2 

°397 

38 

23 

8299 

46o6 

0801 

6881 

2840 

86-72 

4375 

9942 

5368 

o65o 

37 

24 

8572 

4877 

1070 

7i48 

3io4 

8935 

4635 

460200 

5624 

o9o4 

36 

25 

8845 

5i48 

i339 

74i5 

3369 

9i98 

4896 

o458 

588o 

1157 

33 

26 

9n7 

54i8 

1608 

7682 

3634 

946i 

5i56 

o7i6|   6i36 

i4n 

34 

27 

939o 

5689 

1877 

7949 

3899 

9723 

54i7 

o974   6392 

i664 

33 

28 

9662 

596o 

2146 

82i5 

4i64 

9986 

5677 

1232;   6647 

1917 

32 

29 

9935 

623i 

24i5 

8482 

4429 

5937 

i49i:   69o3 

2170 

3i 

3o 

350207 

3665oi 

382683 

398749 

4i4693 

4305xx 

446  i  98 

46i749i  477i59 

49242,4 

?o 

3i 

o48o 

6772 

2952 

9016 

4958 

0774 

6458 

20O7i    74l4 

2677 

29 

32 

0752 

7042 

3221 

9283 

5223 

io36 

67i8 

2265J  767o 

293o 

28 

33 

1025 

73i3 

349o 

9549 

5487 

i299 

6979 

2523   7925!  3i83 

27 

34 

I297 

7584 

3758 

98i6 

5752 

i56i 

7239 

278o!   8i8i|   3436 

26 

35 

i569 

7854 

4027 

400082 

6016 

1823 

7499 

3o38   8436;   3689 

25 

36 

1842 

8i25 

4295 

o349 

6281 

2086 

7759i   3296   8692;   3942 

24 

37 

2114 

8395 

4564 

0616 

6545 

2348 

8019!  3554   8947i  4i95 

23 

38 

2386 

8665 

4832 

0882 

6810 

2610 

8279i   38i2:   92o3 

4448 

22 

39 

2658 

8936 

5ioi 

n49 

7o74 

2873 

8539 

4o69'   9458 

4700 

21 

4o 

35293i 

369206 

385369 

4oi4i5 

4i7338 

433i35 

448799 

464327'4797i3 

494953 

2O 

4i 

32o3 

9476 

5638 

1681 

76o3 

3397 

9o59 

4584 

9968 

5206 

'9 

42 

3475 

9747 

59o6 

i948 

7867 

3659 

93i9 

4842 

480223   5459 

18 

43 

3747 

370017 

6174 

22l4 

8i3i 

392I 

9579 

5ioo 

0479!   5711 

17 

44 

4oi9 

0287 

6443 

2480 

8396 

4i83 

9839 

5357 

0734 

5964 

16 

45 

429I 

o557 

6711 

2747 

8660 

4445 

45oo98 

56i5 

0989 

6217 

i5 

46 

4563 

0828 

6979 

3oi3 

8924 

47o7 

o358 

5872 

1244 

6469 

i4 

47 

4835 

1098 

7247 

3279 

9188 

4969 

0618 

6l29 

1499 

6722 

i3 

48 

5107 

i368 

75iG 

3545 

9452 

523i 

o878 

6387 

1754 

6974 

12 

49 

5379 

i638 

7784 

38n 

97i6 

5493 

ii37 

6644 

2009 

7226 

II 

5o 

35565i 

371908 

388o52 

4o4o78 

4i998o 

435755 

45i397 

4669oi 

482263 

497479 

10 

5i 

5923 

2178 

8320 

4344 

420244 

6oi7 

i656 

7i58 

25i8 

773i 

9 

52 

6i94 

2448 

8588 

46io 

o5o8 

62-78 

I9i6 

74i6 

2773 

7983 

8 

53 

6466 

2718 

8856 

4876 

0772 

654o 

2175 

7673 

3028 

8236 

7 

54 

6738 

2988 

9124 

5i42 

io36 

6802 

2435 

793o 

3282 

8488 

6 

55 

7oio 

3258 

9392 

54o8 

i3oo 

7o63 

2694 

8187 

3537 

874o 

5 

56 

•7281 

3528 

966o 

5673 

i563 

7325 

2953 

8444 

3792 

8992 

A 

57 

7553 

3797 

9928 

5939 

1827 

7587 

32i3 

8701 

4o46 

9244 

3 

58 

7825 

4o67 

39oi96 

62o5 

2O9I 

7848 

3472 

8958 

43oi 

9496 

i 

59 

8o96 

4337 

o463 

647i 

2355 

8110 

373i 

9215 

4555 

9748 

i 

69° 

68° 

67° 

66° 

65° 

64° 

63° 

62° 

61° 

60° 

q 

Natural  Co-sines. 

2 

">5* 

4.5i 

4.48  |  4  45 

4.4i 

4.38 

4.34 

4.3o 

4.26 

4.23  j 

JN  A  T  L  R  A  L    TANGENTS. 


121 


A 

& 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

o 

363970 

383864 

404026 

424475 

445229 

4663o8 

487733 

509525 

531709 

5543o9 

5o 

! 

43oo 

4198 

4365 

48i8 

5577 

6662 

8o93 

9892 

2o83 

4689 

9 

2 

4629 

4532 

4703 

5i62 

5926 

7016 

8453 

5io258 

2456 

5o7o 

8 

3 

4959 

4866 

5o42 

55o5 

6275 

737i 

88i3 

0625 

2829 

545o 

7 

4 

5288 

52OO 

538o 

5849 

6624 

7725 

9i74 

0992 

32o3 

583i 

6 

5 

56i8 

5534 

5719 

6192 

6973 

8080 

9534 

i359 

3577 

6212 

5 

6 

5948 

5868 

6o58 

6536 

7322 

8434 

9895 

I726 

395o 

6593 

4 

7 

6278 

6202 

6397 

6880 

767i 

8789 

490256 

2093 

4324 

6974 

3 

8 

6608 

6536 

6736 

7224 

8020 

9i44 

0617 

2460 

4698 

7355 

2 

9 

6938 

687i 

7o75 

7568 

8369 

9499 

o978 

2828 

5o72 

7736 

I 

10 

367268 

3872o5 

407414 

427912 

4487i9 

469854 

49i339 

DiSigS 

535446 

558ii8 

O 

ii 

7598 

754o 

7753 

8256 

9068 

470209 

1700 

3563 

582i 

8499 

49 

12 

7928 

7874 

8092 

8601 

94i8 

o564 

2061 

SgSo 

6i95 

8881 

48 

i3 

8259 

8209 

8432 

8945 

9768 

0920 

2422 

4298 

657o 

9263 

47 

i4 

8589 

8544 

877i 

9289 

45on7 

I275 

2784 

4666 

6945 

9645 

46 

i5 

8919 

8879 

9111 

9634 

o467 

i63i 

3i45 

5o34 

73i9 

56oo27 

45 

16 

925o 

9214 

945o 

9979 

o8i7 

1986 

35o7 

5402 

7694 

0409 

44 

J7 

958i 

9549 

979° 

43o323 

ii67 

2342 

3869 

577o 

8069 

0791 

:s 

18 

9911 

9884 

4ioi3o 

0668 

i5i7 

2698 

423l 

6i38 

8445 

1174 

.2 

J9 

370242 

39O2I9 

0470 

ioi3 

1868 

3o54 

4593 

65o7 

8820 

i556 

,1 

20 

37o573 

39o554 

4io8io 

43i358 

452218 

4734io 

494955 

5i6875 

539195 

561939 

.O 

21 

o9o4 

0889 

n5o 

I7o3 

2568 

3766 

53i7 

7244 

9571 

2322 

in 

22 

1235 

1225 

1490 

2048 

2919 

4l22 

5679 

76i3 

9946 

2705 

38 

23 

i566 

i56o 

i83o 

2393 

3269 

4478 

6042 

7982 

54o322 

3o88 

37 

24 

1897 

1896 

2I7O 

2739 

3620 

4835 

64o4 

835i 

0698 

347i 

36 

25 

2228 

223l 

25ll 

3o84 

397i 

5191 

6767 

8720 

1074 

3854 

35 

26 

255g 

2567 

285i 

343o 

4322 

5548 

7i3o 

9o89 

i45o 

4238 

34 

27 

289o 

2903 

3192 

3775 

4673 

5go5 

7492 

9458 

1826 

4621 

33 

28 

3222 

3239 

3532 

4l2I 

5o24 

6262 

7855 

9828 

2203 

5oo5 

32 

29 

3553 

3574 

3873 

4467 

5375 

6619 

8218 

52OI97 

2579 

5389 

3i 

3o 

373885 

393910 

4i4ai4 

434812 

455726 

476976 

498582 

52o567 

542956 

565773 

3o 

3t 

4216 

4247 

4554 

5i58 

6o78 

7333 

8945 

o937 

3332 

6i57 

29 

32 

4548 

4583 

4895i  55o4 

6429 

7690 

93o8 

1307 

37o9 

654i 

28 

33 

488o 

4919 

5236J   585o 

678i 

8047 

9672 

1677 

4o86 

6925 

27 

34 

6211 

5255 

5577 

6i97 

7132 

84o5 

5ooo35 

2047 

4463 

73io 

26 

35 

5543 

5592 

59i9 

6543 

7484 

8762 

o399 

2417 

484o 

7694 

25 

36 

5875 

5928 

6260 

6889 

7836 

9120 

o763 

2787 

52i8 

8o79 

24 

37 

6207 

6265 

6601 

7236 

8188 

9^77 

1127 

3i58 

5595 

8464 

23 

38 

6539 

6601 

6943 

7582 

854o 

9835 

1491 

3528 

5973 

8849 

22 

39 

6872 

6938 

7284 

7929 

8892 

480193 

!855 

3899 

635o 

9234 

21 

4o 

377204 

397275 

417626 

438276 

459244 

48o55i 

502219 

524270 

546728 

569619 

2O 

4i 

7536 

7611 

7967 

8622 

9596 

0909 

2583 

464  1 

7106 

57ooo^ 

J9 

42 

7869 

7948 

83o9 

8969 

9949 

1267 

2948 

5OI2 

7484 

oSgo 

1  8 

43 

8201 

8285 

865i 

93i6 

46o3oi 

1626 

33i2 

5383 

7862 

o776 

ll 

44 

8534 

8622 

8993 

9663 

o654 

1984 

3677 

5754 

8240 

1161 

16 

45 

8866 

8960 

9335 

44ooii 

1006 

2343 

4o4i 

6i25 

8619 

i547 

i5 

46 

9199 

9297 

9677 

o358 

i359 

2701 

44o6 

6497 

8997 

i933 

i4 

47 

9532 

9634 

42OOI9 

o7o5 

1712 

3o6o 

4771 

6868 

9376 

2319 

i3 

48 

9864 

9971 

o36i 

io53 

2o65 

34i9 

5i36 

7240 

9755 

27o5 

12 

49 

38019-7 

4oo3o9 

070^ 

i4oo 

2418 

3778 

55o2 

7612 

55oi34 

3092 

II 

5o 

38o53o 

4oo646 

421046 

44i748 

462771 

484i37 

5o5867 

527984 

55o5i3 

573478 

10 

5i 

o863 

0982 

i389 

2095 

3i24 

4496 

6232 

8356 

0892 

3865 

9 

5a 

1196 

1322 

1781 

2443 

3478 

4855 

6598 

8728 

I27I 

4252 

8 

53 

i53o 

1660 

207^ 

279I 

383i 

5212 

6963 

9100 

i65o 

4638 

7 

54 

i863 

1997 

2417 

3i39 

4i85 

5574 

7329 

94?3 

2030 

5026 

6 

55 

2196 

2335 

2759 

3487 

4538 

5933 

7695 

9845 

2409 

54i3 

5 

56 

253o 

£673 

3l02 

3835 

489*2 

6293 

8061 

53o2i8 

2789 

58oo 

4 

57 

2863 

Jon 

3445 

4i83 

5246 

6653 

8427 

oSqi 

3i69 

6187 

3 

58 

3i97 

335o 

3788 

4532 

56oo   701  J 

8793 

096^ 

3549 

6575 

2 

59 

353o 

3688 

4l32 

488o 

5954'   7373 

9i59 

i336 

3929 

6962 

I 

69° 

68° 

67° 

66°   65°  |  64° 

63° 

62° 

61° 

60° 

Natural  Co-tangents. 

* 

P.  r.  5  53 

tol".5'" 

5.6o 

5.68 

5.76 

5.85 

5.95 

6.o5  1  6  16 

f 

6.28 

6.4o  j 

122 


NATURAL    SINES. 


t 

8 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

0 

Sooooo 

5i5o38 

529919 

544639 

559i93 

573576 

587785 

6oi8i5 

6i566i 

62932O 

60 

I 

0252 

5287 

53oi66 

4883 

9434 

38i5 

8021 

2047 

589i 

9546 

59 

2 

o5o4 

5537 

o4i3 

5127 

9675 

4o53 

8256 

2280 

6120 

9772 

58 

3 

0756 

5786 

0659 

5371 

9916 

4291 

8491 

25l2 

6349 

9998 

57 

4 

1007 

6o35 

0906 

56i5 

4529 

8726 

2744 

6578 

630224 

56 

5!   I25g 

6284 

Il52 

5858 

o398 

4767 

8961 

2976 

6807 

o45o 

55 

6 

i5n 

6533 

l399 

6102 

o639 

5oo5 

9196 

3208 

7o36 

0676 

54 

7 

1762 

6782 

1  645 

6346 

0880 

5243 

943i 

344o 

7265 

0902 

53 

8 

2014 

7o3i 

1891 

6589 

II2I 

548i 

9666 

3672 

7494 

1127 

52 

9 

2266 

7280 

2i38 

6833 

i36i 

5719 

9901 

39o4 

7722 

•  i353 

5i 

10 

502517 

5i7529 

532384 

547076 

56i6o2 

575957 

6o4i36 

63i578 

5o 

ii 

2769 

7778 

263o 

7320 

.843 

6195 

0371 

4367 

8180 

1804 

49 

12 

3020 

8027 

2876 

7563 

2o83 

6432 

0606 

4599 

84o8 

2029 

48 

13 

3271 

8276 

3l22 

7807 

2324 

6670 

0840 

483i 

8637 

2255 

47 

i4 

3523 

8525 

3368 

8o5o 

2564 

6908 

1075 

5o62 

8865 

2480 

46 

i5 

3774 

8773 

36i5 

8293 

28o5 

7i45 

i3io 

5294 

9o94 

2705 

45 

16 

4025 

9022 

386i 

8536 

3o45 

7383 

1  544 

5526 

9322 

2931 

44 

17 

4276 

9271 

4io6 

8780 

3286 

7620 

1779 

5757 

955i 

3i56 

43 

18 

4528 

95i9 

4352 

9023 

3526 

7853 

2Ol3 

5988 

9779 

338i 

42 

19 

4779 

9768 

4598 

9266 

3766 

8og5 

2248 

6220 

620007 

36o6 

4i 

20 

5o5o3o 

520016 

534844 

5495o9 

564007 

578332 

592482 

6o645i 

620235 

63383i 

4o 

21 

528i 

0265 

5ogo 

975s 

4247 

857o 

2716 

6682 

o464 

4o56 

39 

22 

5532 

o5i3 

5335 

9995 

4487 

8807 

2g5i 

69i4 

0692 

4281 

38 

23 

5783 

0761 

558i 

55o238 

4727 

9044 

3i85 

7i45 

0920 

45o6 

37 

24 

6o34 

IOIO 

5827 

o48i 

4967 

9281 

3419 

7376 

n48 

4?3i 

36 

25 

6285 

1258 

6072 

0724 

5207 

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3653 

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i376 

4955 

35 

26 

6535 

i5o6 

63i8 

0966 

5447 

9755 

3887 

7838 

1604 

5i8o 

34 

27 

6786 

1754 

6563 

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5687 

9992 

4l2I 

8o69 

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54o5 

33 

28 

7°37 

2002 

6809 

i452 

5927 

580229 

4355 

83oo 

2059 

5629 

32 

29 

7288 

225l 

7o54 

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6166 

o466 

4589 

853i 

2287 

5854 

3i 

3o 

5,07538 

522499 

537300 

55i937 

5664o6 

58o7o3 

594823 

6o876i 

622615 

636o7S 

3o 

3i 

7789 

2747 

7545 

2180 

6646 

0940 

5o57 

8992 

2742 

63o3 

29 

32 

8o4o 

2995 

779° 

2422 

6886 

ii76 

5290 

9223 

2970 

6527 

28 

33 

8290 

3242 

8o35 

2664 

7125 

i4i3 

5524 

9454 

3J97 

675i 

27 

34 

854i 

349o 

8281 

2907 

7365 

i65o 

5758 

9684 

3425 

6976 

26 

35 

8791 

3738 

8526 

7604 

1886 

599i 

99i5 

3652 

7200 

25 

36 

9041 

3986 

8771 

3392 

7844 

2123 

6225 

6ioi45 

388o 

7424 

24 

37 

9292 

4234 

9016 

3634 

8o83 

2359 

6458 

o376 

4io7 

7648 

23 

38 

9542 

448  1 

9261 

3876 

8323 

25g6 

6692 

0606 

4334 

7872 

22 

39 

9792 

4729 

95o6 

4n8 

8562 

2832 

6925 

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456i 

8096 

21 

4o 

5ioo43 

524977 

53975i 

55436o 

5688oi 

533069 

597159 

611067 

624789 

638320 

20 

4i 

0293 

5224 

9996 

4602 

9040 

33o5 

7392 

1297 

5oi6 

8544 

19 

42 

o543 

5472 

4844 

9280 

354i 

7625 

1527 

5243 

8768 

18 

43 

o793 

5719 

o485 

5o86 

95l9 

3777 

7858 

i757 

5470 

8992 

17 

U 

io43 

5967 

0730 

5328 

9758 

4oi4 

8092 

1987 

5697 

9215 

16 

45 

1293 

6214 

0974 

557o 

9997 

425o 

8325 

2217 

5923 

9439 

i5 

46 

1  543 

646  1 

1219 

58i2 

570236 

4486 

8558 

2447 

6i5o 

9663 

i4 

47 

i793 

6709 

i464 

6o54 

0475 

4722 

879I 

2677 

6377 

9886 

i3 

48 

6956 

1708 

6296 

0714 

4g58 

9024 

2907 

66o4 

64ono 

12 

49 

2293 

7203 

i953 

6537 

0952 

5ig4 

9256 

3i37 

683o 

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II 

5o 

5i2543 

52745© 

542197 

556779 

571191 

585429 

599489 

6i3367 

627057 

64o557 

IO 

5i 

2798 

7697 

2442 

7021 

i43o 

5665 

9722 

3596 

7284 

0780 

9 

52 

3o42 

7944 

2686 

7262 

1669 

5901 

9955 

3826 

75io 

ioo3 

8 

53 

3292 

8191 

2930 

75°.4 

1907 

6i37 

600188 

4o56 

7737 

1226 

7 

54 

354i 

8438 

3i74 

7745 

2146 

6372 

O420 

4285 

7963 

i45o 

6 

55 

379i 

8685 

34i9 

7987 

2384 

6608 

o653 

45i5 

8i89 

i673 

5 

56 

4o4o 

8932 

3663 

8228 

2623 

6844 

o885 

4744 

84i6 

1896 

4 

57 

4290 

9179 

3907 

8469 

2861 

7079 

1118 

4974 

8642 

2119 

3 

58 

4539 

9426 

4i5i 

8710 

3ioo 

i35o 

52o3 

8868 

2342 

2 

59 

4789 

9673 

4395 

8952 

3338 

755o 

i583 

5432 

9o94 

2565 

I 

59° 

58° 

57° 

56° 

55° 

54°   53° 

52° 

51° 

50° 

c 

Natural  Co-sines. 

2 

£?>'8 

4.i3 

4-o9 

4-o4 

4.00 

3.95 

3.9o 

3.85  1  3.8o 

3-74  j 

NATURAL    TANGENTS. 


123 


d 

1  s 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

0 

SyySSo 

600861 

624869 

6494o8 

674509 

700208 

726543 

753554 

781286 

8o9784 

60 

I 

7738 

1257 

5274 

982I 

4932 

0641 

6987 

4oio 

1754 

810266 

59 

2 

8126 

i653 

5670. 

650235 

5355 

1075 

7432 

4467 

2223 

o748 

58 

3 

85i4 

2049 

6o83 

o649 

5779 

iSog 

7877 

4923 

2692 

1230 

57 

4 

8903 

2445 

6488 

io63 

6203 

1943 

8322 

538o 

3i6i 

I7I2 

56 

5 

9291 

2842 

6894 

i477 

6627 

2377 

8767 

5837 

363i 

2195 

55 

6 

9680 

3239 

7299 

l892 

7o5i 

2812 

9213 

629.4 

4ioo 

2678 

54 

7 

58oo68 

3635 

77°4 

23o6 

7475 

3246 

9658 

675i 

4570 

3i6i 

53 

8 

0457 

4o32 

8110 

2721 

7900 

368i 

73oio4 

7209 

5o4o 

3644 

52 

9 

o846 

4429 

85i6 

3i36 

8324 

4n6 

o55o 

7667 

55io 

4128 

5i 

10 

58i235 

604827 

628921 

65355i 

678749 

7o455i 

730996 

758i25 

785981 

814612 

5<? 

ii 

1625 

5224 

9327 

3966 

9174 

4987 

i443 

8583 

645  1 

5096 

49 

12 

2Ol4 

5622 

9734 

4382 

9599 

5422 

1889 

9o4i 

6922 

558o 

48. 

i3 

24o3 

6019 

63oi4o 

4797 

680025 

5858 

2336 

95oo 

7394 

6o65 

47 

i4 

2793 

6417 

o546 

52i3 

o45o 

6294 

2783 

9959 

7865 

6549 

46 

1  5 

3i83 

68i5 

0953 

5629 

0876 

673o 

323o 

760418 

8336 

7o34 

45 

16 

3573 

7213 

i36o 

6o45 

1302 

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3678 

0877 

8808 

75i9 

44 

J7 

3963 

7611 

i767 

646  1 

1728 

76o3 

4i25 

i336 

.9280 

8oo5 

43 

18 

4353 

8010 

2I74 

6877 

2i54 

8039 

4573 

1796 

9752 

849i 

4a 

J9 

4743 

84o8 

258i 

7294 

258o 

8476 

5O2I 

2256 

790225 

8976 

4i 

20 

585i34 

608807 

632988 

657710 

683007 

708913 

735469 

762716 

790697 

8i9463 

4o 

21 

5524 

9205 

3396 

8127 

3433 

935o 

5917 

3i76 

1170 

9949 

39 

22 

5gi5 

9604 

38o4 

8544 

386o 

9788 

6366 

3636 

i643 

820435 

38 

23 

63o6 

6iooo3 

4211 

8961 

4287 

710225 

68i5 

4o97 

2117 

O922 

37 

24 

6697 

o4o3 

4619 

9379 

47i4 

o663 

7264 

4558 

2590 

i4o9 

36 

25 

7088 

0802 

5o27 

9796 

6142 

IIOI 

77i3 

5019 

3o64 

i897 

35 

26 

7479 

I2OI 

5436 

660214 

5569 

1539 

8162 

548o 

3538 

2384 

34 

27 

7870 

1601 

5844 

o63i 

5997 

i977 

8611 

594i 

4OI2 

2872 

33 

28 

8262 

2OOI 

6253 

1049 

6425 

2416 

9061 

64o3 

4486 

336o 

32 

29 

8653 

2401 

6661 

1467 

6853 

2854 

95ii 

6865 

4961 

3848 

3i 

3o 

589045 

612801 

637070 

661886 

687281 

•713293 

73996i 

767327 

795436 

824336 

3o 

3i 

9437 

3201 

7479 

23o4 

7709 

3732 

74o4n 

7789 

5911 

4825 

29 

32 

9829 

36oi 

7888 

2723 

8i38 

4i7i 

0862 

8252 

6386 

53i4 

28 

33 

59022I 

4OO2 

8298 

3i4i 

8567 

46n 

l3l2 

8714 

6862 

58o3 

27 

34 

o6i3 

4402 

8707 

356o 

8995 

5o5o 

i763 

9177 

?337 

6292 

26 

35 

1006 

48o3 

9117 

3979 

94  2  5 

5490 

22l4 

9640 

7813 

6782 

25 

36 

!398 

52o4 

95a7 

4398 

9854 

593o 

2666 

770104 

8290 

7272 

24 

3? 

i79i 

56o5 

9937 

48i8 

690283 

637o 

3u7 

o567 

8766 

7762 

23 

38 

2184 

6006 

64o347 

5237 

0713 

6810 

3569 

io3i 

9242 

8252 

22 

39 

2577 

64o8 

o757 

5657 

n43 

725o 

4O2O 

i4g5 

97i9 

8743 

21 

4o 

592970 

616809 

64n67 

666o77 

69i572 

7i769i 

744472 

771959 

800196 

820.234 

20 

4i 

3363 

7211 

i578 

6497 

2OO3 

8i32 

4925 

2423 

0674 

9725 

T9 

42 

3757 

76i3 

i989 

6917 

2433 

8573 

5377 

2888 

ii5i 

83o2i6 

18 

43 

4i5o 

8oi5 

2399 

7337 

2863 

9014 

583o 

3353 

1629 

0707 

I7 

44 

4544 

84i7 

2810 

7758 

3294 

9455 

6282 

38!8 

2107 

"99 

16 

45 

4937 

88i9 

3222 

8i79 

3725 

9897 

6735 

4283 

2585 

i69i 

i5 

46 

533i 

922I 

3633 

8599 

4i56 

72o339 

7189 

4748 

3o63 

2i83 

i4 

47 

5725 

9624 

4o44 

9020 

4587 

o78i 

7642 

52i4 

3542 

2676 

i3 

48 

6120 

62OO26 

4456 

9442 

5oi8 

1223 

8096 

568o 

4021 

3i69 

12 

49 

65i4 

0429 

4868 

9863 

545o 

i665 

8549 

6i46 

45oo 

3662 

II 

5o 

5969o8 

620832 

64528o 

670284 

69588i 

722IO8 

749oo3 

776612 

804979 

834i55 

IO 

5i 

73o3 

1235 

5692 

0706 

63i3 

255o 

9458 

7078 

5458 

4648 

9 

52 

7698 

i638 

6io4 

1128 

6745 

2993 

9912 

7545 

5938 

5i42 

8 

53 

8o93 

2042 

65i6 

i55o 

7177 

3436 

75o366 

8012 

64i8 

5636 

7 

54 

8488 

2445 

6929 

I972 

7610 

3879 

0821 

8479 

6898 

6i3o 

6 

55 

8883 

2849 

7342 

2394 

8042 

4323 

1276 

8946 

7379 

6624 

5 

56 

0278 

3253 

7755 

2817 

8475 

4766 

i73i 

94i4 

7859 

7119 

4 

57 

9.674 

3657 

8168 

324o 

89o8 

52IO 

2187 

9881 

834o 

7614 

3 

58 

600069 

4o6i 

858i 

3662 

934i 

5654 

2642 

780349!  8821 

8109 

2 

59 

o465 

4465 

8994 

4o85 

9774 

6o98 

3o98 

0817)   93o3 

86o4 

I 

59° 

58°  J  57° 

56° 

55° 

54° 

53° 

52° 

51° 

50° 

d 

Natural  Co-tangents. 

1 

P.P.-  „ 
to  I".6'53 

6.67 

6.82 

6.97 

7.14 

7.3i 

7-5o  |  7.70 

7.92 

8.  .4 

124 


NATURAL    SINES. 


J) 

• 

40° 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

0 

642788 

656o59 

669131 

681998 

694658 

707107 

719340 

73i354 

743i45 

754710 

60 

I 

3oio 

6279 

9347 

2211 

4868 

73l2 

9542 

i552 

3339 

49oo 

5o 

2 

3233 

6498 

9563 

2424 

5077 

75i8 

9744 

i75o 

3534 

5oqi 

58 

3 

3456 

6717 

9779 

2636 

5286 

7724 

9946 

i949 

3728 

5282 

57 

4 

3679 

6937 

9995 

2849 

5495 

7929 

720148 

2l47 

3923 

5472 

56 

5 

3901 

7i56 

670211 

3o6i 

5704 

8i34 

0349 

2345 

4ii7 

5663 

55 

6 

4124 

7375 

0427 

3274 

59i3 

834o 

o55i 

2543 

4312 

5853 

54 

7 

4346 

7594 

0642 

3486 

6122 

8545 

0753 

274l 

45o6 

6o4^ 

53 

8 

4569 

7814 

o858 

3698 

633o 

875o 

o954 

2939 

47oo 

6234 

52 

9 

4791 

8o33 

107^ 

3911 

6539 

8956 

11.56 

3i37 

4894 

6425 

5i 

10 

645oi3 

658252 

671289 

684i23 

696748 

7o9i6i 

721357 

733334 

745o88 

7566i5 

5o 

ii 

5236 

8471 

i5o5 

4335 

6957 

9366 

i559 

3532 

5282 

68o5 

49 

12 

5458 

8689 

1721 

4547 

7i65 

957i 

1760 

373o 

5476 

6995 

48 

i3 

568o 

8908 

i936 

4759 

7374 

9776 

1962 

3927 

567o 

7i85 

47 

i4 

5902 

9127 

2l5l 

4971 

7582 

9981 

2i63 

4l25 

5864 

7375 

46 

i5 

6124 

9346 

2367 

5i83 

779° 

710185 

2364 

4323 

6057 

7565 

45 

16 

6346 

9565 

2582 

5395 

7999 

0390 

2565 

4520 

625i 

7755 

44 

17 

6568 

9783 

2797 

56o7 

8207 

o595 

2766 

47i7 

6445 

7945 

43 

18 

6790 

660002 

3oi3 

58i8 

84i5 

0799 

2967 

4gi5 

6638 

8:34 

42 

'9 

7012 

O22O 

3228 

6o3o 

8623 

ioo4 

3  1  68 

5lI2 

6832 

8324 

4i 

20 

647233 

660439 

673443 

686242 

698832 

711209 

723360 

7353o9 

747025 

7585i4 

4o 

21 

?455 

0657 

3658 

6453 

9o4o 

i4i3 

357o 

55o6 

7218 

87o3 

39 

22 

7677 

0875 

3873 

6665 

9248 

1617 

377i 

57o3 

74l2 

8893 

38 

23 

7898 

1094 

4o88 

6876 

9455 

1822 

397i 

59oo 

76o5 

9082 

37 

24 

8120 

l3l2 

4302 

7088 

9663 

2026 

4l72 

6o97 

7798 

927I 

36 

26 

834i 

i53o 

45i7 

7299 

9871 

2230 

4372 

6294 

7991 

946i 

35 

26 

8563 

1748 

4732 

75io 

700079 

2434 

4573 

649i 

8i84 

965o 

34 

27 

8784 

1966 

4947 

7721 

0287 

2639 

4773 

6687 

8377 

9839 

33 

28 

9006 

2184 

5i6i 

7932 

0494 

2843 

4974 

6884 

857o 

760028 

32 

29 

9227 

2402 

5376 

8x44 

0702 

3o47 

5i74 

7o8i 

8763 

0217 

3i 

3o 

649448 

662620 

675590 

688355 

700909 

7i325o 

725374 

737277 

748956 

76o4o6 

3o 

3i 

9669 

2838 

58o5 

8566 

1117 

3454 

5575 

7474 

9i48 

o595 

29 

32 

9890 

3o56 

6019 

8776 

i324 

3658 

5775 

767o 

934i 

0784 

28 

33 

65om 

3273 

6233 

8987 

i53i 

3862 

5975 

7867 

9534 

o97a 

27 

34 

o332 

349i 

6448 

9198 

i739 

4o66 

6175 

8o63 

9726 

1161 

26 

35 

o553 

37o9 

6662 

9409 

i946 

4269 

6375 

8259 

99i9 

i35o 

25 

36 

0774 

3926 

6876 

9620 

2i53 

4473 

6575 

8455 

75oin 

i538 

24 

37 

o995 

4i44 

7090 

983o 

236o 

4676 

6775 

865i 

o3o3 

1727 

23 

38 

1216 

436i 

73o4 

690041 

2567 

488o 

69  74 

8848 

o496 

i9i5 

22 

39 

i437 

4579 

75i8 

025l 

2774 

5o83 

7i74 

9o43 

0688 

2IO4 

21 

4o 

**z657 

664796 

677732 

690462 

7O298i 

715286 

727374 

739239 

75o88o 

762292 

2O 

4i 

1878 

5oi3 

7946 

0672 

3i88 

5490 

7573 

9435 

I072 

2480 

!9 

4s 

2098 

523o 

8160 

0882 

3395 

5693 

7773 

963i 

1264 

2668 

18 

43 

2319 

5448 

8373 

io93 

36oi 

5896 

7972 

9827 

i456 

2856 

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44 

a539 

5665 

8587 

i3o3 

38o8 

6099 

8l72 

740023 

1  648 

3o44 

16 

45 

2760 

5882 

8801 

i5i3 

4oi5 

63o2 

837i 

0218 

1840 

3232 

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46 

2980 

6099 

9014 

1723 

4221 

65o5 

857o 

o4i4 

2032 

3420 

i4 

4? 

32OO 

63i6 

9228 

i933 

4428 

6708 

8769 

o6o9 

2223 

36o8 

i3 

48 

3421 

6532 

944  1 

2143 

4634 

6911 

8969 

o8o5 

24i5 

3796 

12 

49 

364i 

6749 

9655 

2353 

484i 

7n3 

9168 

IOOO 

2606 

3984 

II 

5o 

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666966 

679868 

692565 

7o5o47 

7T73i6 

729367 

74n95 

752?98 

764171 

10 

5i 

4o8i 

7i83 

680081 

2773 

5253 

75l9 

9566 

i39i 

2989 

4359 

9 

62 

43oi 

7399 

0295 

2983 

5459 

772i 

9765 

i586 

3i8i 

4547 

8 

53 

452i 

7616 

o5o8 

3l92 

5665 

7924 

9963 

1781 

3372 

4734 

7 

54 

4?4i 

7833 

0721 

3402 

5872 

8126  73oi62 

i976 

3563 

492I 

6 

55 

4961 

8o49 

o934 

36n 

6078 

832g!   o36i 

2I7I 

3755 

5io9 

5 

56 

5i8o 

8265 

n47 

382i 

6284 

853i 

o56o 

2366 

3946 

5296 

4 

£z 

54oo 

8482 

i36o 

4o3o 

6489 

8733 

o758 

256i 

4i37 

5483 

3 

58 

562O 

8698 

i573 

424o 

6695 

8936 

o957 

2755 

4328 

567o 

2. 

59 

5839 

8914 

1786 

4449 

6901 

9i38 

u55 

295o 

45i9 

5857 

I 

49° 

48° 

47° 

46° 

45° 

44° 

43° 

42° 

41° 

40° 

A 

Natural  Co-sines. 

S 

f0-£3.69 

S.63 

3.57 

3.52 

3.46 

3.4o 

3.34 

3.27 

3.2i  j  3.i5 

NATURAL    TANGENTS. 


d 

H 

40° 

41°  !  42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

O 

839ioo 

869287  9oo4o4 

o325i5 

965689 

I.OOOOO 

i.o3553 

1.07237 

1.  11061 

i.i5o37 

60 

I 

9595 

979s 

ogSi!   3o5g 

625i 

oo58 

36i3 

7299 

1126 

5ic4 

59 

2 

84oo92 

i458 

36o3 

68i4 

0116 

3674 

7362 

n9i 

5172 

58 

3 

o588 

0820 

i985 

4i48 

7377 

0175 

3734 

7425 

1256 

524.0 

57 

4 

1084 

i332 

25i3 

4693 

0233 

3794 

7487 

1321 

53o8 

56 

5 

i58i 

i843 

3o4i 

5238 

85o4 

O29I 

3855 

755o 

1387 

5375 

55 

6 

2078 

2356 

3569 

5783 

9067 

o35o 

39i5 

76i3 

i452 

5443 

54 

7 

2575 

2868 

4098 

6329 

9632 

o4o8 

3976 

7676 

i5i7 

55n 

53 

8 

3073 

338i 

4627 

6875 

970196 

0467 

4o36 

7738 

i582 

5579 

52 

9 

357i 

3894 

5i56 

7422 

0761 

o525 

4097 

7801 

1  648 

5647 

5i 

10 

844o69 

874407 

9o5685 

937968 

971326 

i.oo583 

I.o4i58 

1.07864 

1.11713 

1.  16715 

5o 

ii 

4567 

492O 

62i5 

85i5 

1892 

0642 

4218 

7927 

1778 

5783 

49 

12 

5o66 

5434 

6745 

9063 

2458 

0701 

4279 

799° 

1  844 

585i 

48 

i3 

5564 

5948 

7275 

9610 

3024 

°759 

434o 

8o53 

J9°9 

5919 

47 

i4 

6o63 

6462 

78o5 

94oi58 

359o 

0818 

44oi 

8116 

i975 

5987 

46 

i5 

6562 

6076 

8336 

0706 

4i57 

0876 

446  1 

8179 

2041 

6o56 

45 

16 

7062 

7491 

8867 

1255 

4724 

o935 

4522 

8243 

2106 

6124 

44 

17 

7662 

8006 

9398 

i8o3 

5291 

o994 

4583 

83o6 

2I72 

6192 

43 

18 

8062 

852i 

993o 

2352 

5859 

io53 

4644 

8369 

2238 

6261 

42 

19 

8562 

9o37 

910462 

2902 

6427 

III2 

47o5 

8432 

2303 

6329 

4i 

20 

849,062 

870.553 

910994 

94345i 

976996 

I.OII7O 

1.04766 

i.o8496 

i.i2369 

1.16398 

4o 

2! 

9563 

880060. 

i526 

4ooi 

7564 

I229 

4827 

8559 

2435 

6466 

39 

22 

85ooG4 

o585 

2059 

4552 

8i33 

1288 

4888 

8622 

25oi 

6535 

38 

23 

o565 

I  102 

2592 

5  1  02 

87o3 

1  347 

4949 

8686 

2567 

66o3 

37 

24 

1067 

i6i9 

3i25 

5653 

9272 

i4o6 

5oio 

8749 

2633 

6672 

36 

25 

i568 

2i36 

3659 

8204 

i465 

5072 

88i3 

2699 

6741 

35 

26 

2070 

2653 

4193 

6756 

980413 

i524 

5i33j   8876 

2765 

6809 

34 

27 

2573 

3i7i 

4727 

73o7 

0983 

i583 

5i94 

894o 

283i 

6878 

33 

28 

3o75 

3689 

5261 

7859 

1  554 

1  642 

5255 

9oo3 

2807 

6947 

32 

29 

3578 

4207 

5796 

8412 

2126 

1702 

53i7 

9o67 

2963 

7016 

3i 

3o 

854o8i 

884725 

9i633i 

948965 

982697 

1.01761 

1.05378 

i.o9i3i 

I.i3o29 

1.17085 

3o 

3i 

4584 

5244 

6866 

95i8 

3269 

1820 

5439 

9i95 

3o96 

7i54 

29 

32 

5o87 

5763 

7402 

95oo7i 

3842 

1879 

55oi 

9258 

3i62 

7223 

28 

33 

559i 

6282 

7938 

0624 

44i4 

I939 

5562 

9322 

3228 

7292 

27 

34 

6o95 

6802 

8474 

1178 

4987 

i998 

5624 

9386 

3295 

736i 

26 

35 

6599 

7321 

9010 

1733 

556o 

2057 

5685 

945o 

336i 

743o 

25 

36 

7104 

7842 

9547 

2287 

6i34 

2117 

5747 

95i4 

3428 

75oo 

24 

37 

7608 

8362 

920084 

2842 

6708 

2176 

5809 

9578 

3494 

7569 

23 

38 

8n3 

8882 

0621 

3397 

7282 

2236 

587o 

356i 

7638 

22 

39 

86i9 

V4o3 

n59 

SgSS 

7867 

2295 

5932 

97o6 

3627 

7708 

21 

4o 

859i24 

889924 

921697 

9545o8 

988432 

1.02355 

i.o5994 

1.13694 

1.17777 

2O 

4i 

963o 

890446 

2235 

5o64 

9007 

24i4 

6o56 

9834 

3761 

7846 

19 

42 

86oi36 

0967'   2773 

562i 

9582 

2474 

6117 

9899 

3828 

7916 

18 

43 

0642 

1489!   33i2 

6177 

99oi58 

2533 

6179 

n  o 

0963 

3894 

7986 

i7 

441  n48 

2012 

385i 

6734 

o735 

2593 

6241 

1.10027 

3961 

8o55 

16 

45 

i655 

2534 

439o 

7292 

i3n 

2653 

63o3 

0091 

4028 

8i25 

i5 

46 

2162 

3o57 

493o 

7849 

1888 

2713 

6365 

oi56 

4095 

8194 

i4 

47 

2669 

358o 

5470 

84o7 

2465 

2772 

6427 

O22O 

4162 

8264 

i3 

48 

3i77 

4io3 

6oioj   8966 

3o43 

2832 

6489 

0285 

4229 

8334 

12 

49 

3685 

4627 

655i 

9524 

362i 

280.2 

655i 

o349 

4296 

84o4 

II 

5o 

864i93 

895i5i 

927091 

960083 

994199 

1.02962 

i.o66i3 

i.io4i4 

1.  14363 

i.i8474 

IO 

5i 

4701 

5675i   7632 

0642 

4778 

3012 

6676 

o478 

443o 

.8544 

9 

52 

5209 

6199   8i74 

1202 

5357 

3072 

6738 

o543 

4498 

86i4 

8 

53 

57i8 

67241  87i5 

1761 

5936 

3i32 

6800 

o6o7 

4565 

8684 

7 

54 

6227 

•7249!  9257 

2322 

65i5 

3l92 

6862 

o672 

4632 

8754 

6 

55 

6736 

7774 

9800 

2882 

7°95 

3252 

6925 

o737 

4699 

8824 

5 

56 

7246 

8299 

93o342 

3443 

7676 

33i2 

6987 

0802 

4767 

8894 

4 

57 

7756 

8825 

o885 

4oo^ 

8256 

3372 

7049 

o867 

4834 

8964 

3 

58 

8266 

935i 

1428 

4565 

8837 

3433 

7112 

0931 

4902 

9o35 

2 

59 

8776 

9877 

1971 

5l27 

94i8 

3493 

7i74 

o996 

4969 

9io5 

I 

49° 

48° 

47° 

46° 

45° 

44° 

43° 

42°   41° 

40° 

d 

Natural  Co-tangents. 

•H 

P.  P. 

8.64   8.92   9  2i 

9.53  1  o.99 

i  .02 

i.  06 

I.  10 

,.,5 

.  _J 

126 


A  T  U  R  A  L 


S 


NES. 


1 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

o 

766044 

777146 

788011 

798636 

8o9oi7 

8i9i52 

829o38 

83867i 

848o48 

857i67 

60 

I 

623i 

7329 

8190 

8811 

9188 

93i9 

9200 

8829 

8203 

73i7 

59 

2 

64i8 

7512 

8369 

8985 

9359 

9486 

9363 

8987 

8356 

7467 

58 

3 

66o5 

.  7695 

8548 

9160 

953o 

9652 

9525 

9146 

85io 

7616 

57 

4 

6792 

7878 

8727 

9335 

97oo 

9819 

9688 

93o4 

8664 

7766 

56 

F 

6979 

8060 

89o5 

95io 

987i 

9985 

985o 

9462 

8818 

79i5 

55 

6 

7i65 

8243 

9084 

9685 

810042 

82Ol52 

83ooi2 

9620 

8972 

8o65 

54 

7 

7352 

8426   9263 

9859 

O2I2 

o3i8 

oi  74 

9778 

9125 

8214 

53 

8 

7538 

86o8J   944i 

8ooo34 

o383 

o485 

o337 

9g36 

9279 

8364 

52 

9 

7725 

879i 

9620 

0208 

o553 

o65i'  o499 

840094 

9433 

85i3 

5i 

10 

767911 

778973 

789798 

8oo383 

8io723 

820817'  83o66i 

84025  i 

849586 

858662 

5o 

ii 

8097 

9i56 

9977 

o557 

o894 

o983 

0823 

o4o9 

9739 

8811 

49 

12 

8284 

9338 

79oi55 

0731 

1064 

n49 

0984 

o567 

9893 

896o 

48 

i3 

8470 

952O 

o333 

0906 

1234 

i3i5 

n46 

0724 

85oo46 

9I09 

47 

x4 

8656 

9702 

o5n 

1080 

i4o4 

i48i 

i3o8 

0882 

or99 

9258 

46 

i5 

8842 

9884 

o69o 

1254 

i574 

i647 

i47o 

io39 

o352 

94o6 

45 

16 

9028 

780067 

0868 

1428 

i744!   i8i3 

i63i 

1196 

o5o5 

9555 

44 

*7 

9214 

0249 

1046 

1602 

i9i4 

1978 

I793 

1  354 

o658 

97°4 

43 

x8 

9400 

o43o 

1224 

1776 

2084 

2i44 

i954 

i5n 

0811 

9S52 

42 

J9 

9585 

0612 

i4oi 

1949 

2253 

23lO 

2Il5 

1668 

o964 

860001 

4i 

20 

769771 

7807^ 

79l579 

802123 

812423 

822475 

832277 

84i8s5 

85ni7 

86oi49 

4o 

21 

9957 

o976 

1757 

2297 

2592 

264l:    2438 

1982 

I269 

0297 

39 

22 

770142 

n57 

'  i935 

2470 

276>2 

2806 

2599 

2i39 

1422 

o446 

38 

23 

o328 

i339 

2112 

2644 

293i 

2971 

2760 

2296 

1575 

o594 

37 

24 

o5i3 

l520 

2290 

281-7 

3ioi 

3i36 

292I 

2452 

1727 

0742 

36 

25 

0699 

1702 

2467 

2991 

327O 

33o2 

3o82 

2609 

i879 

o89o 

35 

26 

0884 

i883 

2644 

3i64 

3439 

3467 

3243 

2766 

2O32 

io38 

34 

27 

1069 

2o65 

2822 

3337 

36o8 

3632 

34o4 

2922 

2184 

1186 

33 

28 

1254 

2246 

2999 

35n 

3778 

3797 

3565 

3°79 

2336 

i334 

32 

29 

i44o 

2427 

3176 

3684 

3947 

396i 

3725 

3235 

2488 

i48i 

3i 

3o 

771625 

782608 

793353 

8o3857 

8i4n6 

824126 

833886 

84339i 

85264o 

86i629 

3o 

3i 

1810 

2789 

353o 

4o3o 

4284 

4291 

4o46 

3548 

2792 

i777 

29 

32 

i995 

2970 

37o7 

42o3 

4453 

4456 

4207 

3704 

2944 

I924 

28 

33 

2179 

3i5i 

3884 

4376 

4622 

4620 

4367 

386o 

3096 

2072 

27 

34 

2364 

3332 

4o6i 

4548 

4791 

4785 

4527 

4oi6 

3248 

22I9 

26 

35 

2549 

35i3 

4238 

472I 

4959 

4949 

4688 

4172 

3399 

2366 

25 

36 

2734 

3693 

44i5 

4894 

5i28 

5n3 

4848 

4328 

355i 

25i4 

24 

37 

2918 

3874 

459i 

5o66 

5296 

5278 

5  008 

4484 

3702 

2661 

23 

38 

3io3 

4o55 

4768 

5239 

5465 

5442 

5i68 

464o 

3854 

2808 

22 

39 

3287 

4235 

4944 

54n 

5633 

56o6 

5328 

4795 

4oo5 

2955 

21 

4o 

773472 

784416 

795121 

8o5584 

8i58oi 

825770 

835488 

84495i 

854i56 

863io2 

2O 

4i 

3656 

4596 

5297 

5756 

5969 

5934 

5648 

5io6 

43o8 

3249 

19 

42 

384o 

4776 

5473 

5928 

6!38 

6098 

58o7 

5262 

4459 

3396 

18 

43 

4024 

4957 

565o 

6100 

63o6 

6262 

5967 

54  1  7 

46io 

3542 

ll 

44 

4209 

5i37 

5826 

6273 

6474 

6426 

6127 

5573 

476i 

3689 

16 

45 

4393 

53i7 

6002 

6445 

6642 

659o 

6286 

5728 

49I2 

3836 

i5 

46 

4577 

5497 

6i78 

66i7 

6809!   6753 

6446 

5883 

5o63 

3982 

i4 

4? 

4761 

5677 

6354 

6788 

6977!   69i7 

66o5 

6o38 

52I4 

4128 

i3 

48 

4944 

5857 

653o 

6960 

7i45.   7o8i 

6764 

6i93 

5364 

4275 

12 

49 

5i28 

6o37 

1  67o6 

7l32 

73i3|   7244 

6924 

6348 

55!5 

442i 

II 

5o 

7753i2 

786217 

796882 

8o73o4 

817480  827407 

837o83 

8465o3 

855665 

864567 

IO 

5i 

5496 

6396 

7o57 

7475 

7648 

7571 

7242 

6658 

58i6 

47i3 

9 

52 

5679 

6576 

7233 

7647 

78i5 

7734 

74oi 

68;3 

5966 

486o 

8 

53 

5863 

6756 

74o8 

7818 

7982 

7897 

756o 

6967 

6ii7 

5oo6 

7 

54 

6o46 

6.935 

7584 

799° 

8i5o 

8060 

7719 

7I22 

6267 

5i5i 

6 

55 

623o 

7114 

7759 

8161 

83i7 

8223 

7878 

7277 

64i7 

5297 

5 

56 

64i3 

7294 

7935 

8333 

8484 

8386 

8o36 

743i 

6567 

5443 

4 

57 

6596 

7473 

8110 

85o4 

865i 

8549 

8i95 

7585 

67i8 

5589 

3 

58 

6780 

7652 

8285 

8675 

8818 

8712 

8354 

774o 

6868 

5734 

2 

59 

6963 

7832 

846o 

8846 

8985 

8875 

85i2 

7894 

7017 

588o 

I 

39° 

38° 

37° 

.36° 

35° 

34° 

33° 

32° 

31° 

30° 

£3 

Natural  Co-sines. 

•9 

3 

££3.o8 

3.02 

2.95   2.88 

2.81   2.75 

2.68 

2.60 

2.53 

2.46 

NATURAL    TANGENTS. 


A 

i 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

0 

1.19176 

1.23490 

1.27994 

i.327o4 

i.37638 

1.42816 

1.48266 

1.53986 

i.6oo33 

1.66428 

60 

i 

9246 

3563 

8071 

Q785 

7722 

2903 

8349 

4o85 

oi37 

6538 

59 

2 

93i6 

3637 

8i48 

2865 

78o7 

2992 

8442 

4i83 

0241 

6647 

58 

3 

9387 

37io 

8226 

2946 

789i 

3o8o 

8536 

4281 

o345 

6767 

57 

4 

9457 

3784 

8302 

3026 

7976 

3169 

8629 

4379 

o449 

6867 

56 

5 

9628 

3858 

8379 

3io7 

8060 

3258 

8722 

44?8 

o553 

6978 

55 

6 

9699 

SgSi 

8456 

3i87 

8i45 

3347 

88l6 

4576 

o657 

7088 

54 

7 

9660 

4oo5 

8533 

3268 

8229 

3436 

8909 

4675 

o76i 

7198 

53 

8 

f  .' 
974o 

4o79 

8610 

3349 

83i4 

3525 

9OO3 

4774 

0866 

73°9 

62 

9 

9811 

4i53 

8687 

343o 

8399 

36i4 

9°97 

4873 

o97o 

?4*v 

5i 

Zo 

1.19882 

1.24227 

1.28764 

i.335ii 

1.38484 

1.43703 

1.49190 

1.54972 

i.6io74 

1.67630 

5o 

II 

9953 

43oi 

8842 

3592 

8568 

3792 

9284 

6071 

1179 

764i 

49 

12 

1.20024 

4375 

8919 

3673 

8653 

388i 

9378 

6170 

1283 

7752 

48 

i3 

0096 

4449 

8997 

3754 

8738 

397° 

94-72 

6269 

i388 

7863 

47 

i4 

0166 

4523 

9o74 

3835 

8824 

4o6o 

9666 

5368 

i4g3 

7974 

46 

i5 

0237 

4597 

9162 

39i6 

89o9 

4i49 

9661 

5467 

1698 

8o85 

45 

16 

o3o8 

4672 

9229 

3998 

8994 

4239 

9755 

6667 

I7o3 

8196 

44 

i7 

o379 

4746 

93o7 

4o79 

9°79 

4329 

9849 

5666 

1808 

83o8 

43 

18 

o45i 

4820 

9385 

4i6o 

9i65 

44i8 

9944 

6766 

1914 

8419 

42 

J9 

O522 

4895 

9463 

4242 

925o 

45o8 

i.5oo38 

5866 

2019 

853i 

4i 

20 

1.20693 

1.24969 

i.2954i 

1.34323 

i.39336 

1.44598 

i.5oi33 

1.66966 

1.62126 

1.68643 

4o 

21 

o665 

5o44 

96i8 

44o5 

942I 

4688 

0228 

6o65 

223O 

8754 

39 

22 

o736 

6118 

9696 

4487 

95o7 

4778 

0322 

6i65 

2336 

8866 

38 

23 

0808 

5i93 

9775 

4568 

9593 

4868 

0417 

6266 

2442 

8979 

37 

24 

08  79 

6268 

9853 

465o 

9679 

4958 

0612 

6366 

2648 

9091 

36 

25 

0961 

5343 

993i 

4732 

9764 

5o49 

0607 

6466 

2664 

9203 

35 

26 

1023 

6417 

1.30009 

48i4 

9860 

5i39 

0702 

6566 

276o 

9316 

34 

27 

1094 

5492 

008  7 

4896 

9936 

5229 

0797 

6667 

2866 

9428 

33 

28 

1166 

5567 

0166 

4978 

1.40022 

5320 

o893 

6767 

2972 

9641 

32 

29 

1238 

6642 

0244 

5o6o 

OIO9 

54io 

0988 

6868 

3o79 

9653 

3i 

3o 

I.2l3lO 

1.26717 

i.3o323 

i.35i42 

i.4oi95 

i.455oi 

i.  61084 

1.66969 

i.63i85 

i.69766 

3o 

3i 

i382 

679,2 

o4oi 

6224 

0281 

5592 

1179 

7069 

3292 

9879 

29 

32 

i454 

6867 

o48o 

5307 

o367 

6682 

1276 

7170 

3398 

9992 

28 

33 

1626 

5943 

o558 

5389 

o454 

5773 

1370 

7271 

35o5 

1.70106 

27 

34 

1698 

6018 

o637 

6472 

o54o 

5864 

i486 

7372 

36i2 

0219 

26 

35 

1670 

6o93 

0716 

5554 

o627 

5955 

1662 

7474 

37i9 

o332 

26  i 

36 

1742 

6i69 

0796 

5637 

0714 

6o46 

1668 

7575 

3826 

o446 

24 

37 

1814 

6244 

0873 

6719 

0800 

6i37 

i754 

7676 

3934 

0660 

23 

38 

1886 

63i9 

0962 

6802 

0887 

6229 

1860 

7778 

4o4i 

o673 

22 

39 

1969 

6395 

io3i 

5885 

o974 

6320 

1946 

7879 

4i48 

o787 

21 

4o 

I.22o3l 

1.26471 

i.Sino 

i.35968 

i.4io6i 

1.45411 

1.62043 

1.67981 

1.64266 

20 

4i 

2IO4 

6546 

1190 

6o5i 

n48 

65o3 

2i3g 

8o83 

4363 

1016 

19 

42 

2176 

6622 

1269 

6i34 

1235 

6595 

2235 

8i84 

447i 

1129 

18 

43 

2249 

6698 

1  348 

6217 

1322 

6686 

2332 

8286 

4579 

1244 

i7 

44 

2321 

6774 

1427 

63oo 

i4o9 

6778 

2429 

8388 

4687 

i358 

16 

45 

2394 

6849 

1607 

6383 

i497 

687o 

2626 

8490 

4795 

i473 

i5 

46 

2467 

6925 

1686 

6466 

1  584 

6962 

2622 

8593 

4903 

1  588 

i4 

47 

2539 

7001 

1666 

6649 

l672 

7o53 

2719 

8696 

6011 

I702 

i3 

48 

2612 

7°77 

1745 

6633 

i759 

7i46 

2816 

8797 

6120 

1817 

12 

49 

2686 

7i53 

1826 

6716 

•  i847 

7238 

2913 

8900 

6228 

1932 

II 

5o 

1.22768 

I.2723o 

i.3i9o4 

i.368oo 

1.41934 

i.4733o 

i.53oio 

1.69002 

1.65337 

1.72047 

IO 

5i 

283i 

73o6 

1984 

6883 

2O22 

7422 

3107 

9106 

5445 

2i63 

9 

62 

2904 

7382 

2064 

6967 

2110 

75i4 

32o5 

9208 

5554 

2278 

8 

53 

2977 

7458 

2i44 

7060 

2198 

76o7 

3302 

93n 

5663 

2393 

7 

54 

3o5o 

7535 

2224 

7i34 

2286 

7699 

34oo 

94i4 

6772 

2609 

6 

55 

3i23 

76n 

23o4 

7218 

2374 

7792 

3497 

95l7 

6881 

2626 

5 

56 

3196 

7688 

2384 

7302 

2462 

7885 

3595 

9620 

6990 

274l 

4 

57 

3270 

7764 

2464 

7386 

2660 

7977 

3693 

9723 

6099 

2857 

3 

5b 

3343 

784i 

2544 

747o 

2638 

8o7o 

379i 

9826 

6209 

2973 

2 

69 

34i6 

79r7 

2624 

7554 

2726 

8i63 

3888 

9930 

63i8 

3o89 

I 

39° 

38°   37° 

36° 

35° 

34° 

33° 

32° 

31° 

30° 

a 

Natural  Co-tangents. 

fo'i">ao 

1.26 

i.3i 

i.37 

i.44 

1.5, 

!.59 

1.68 

1.78 

1.88  i 

128 


NATURAL    SINES. 


4 

j 

GO0 

61° 

62° 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

o 

866025 

874620 

882948 

891007 

898794 

906308 

9i3545 

92o5o5 

92718^ 

93358o 

60 

I 

6171 

4761 

3o84 

1139 

8922 

643i 

3664 

0618 

7293 

3685 

59 

2 

63i6 

4902 

3221 

1270 

9049 

6554 

3782 

0732 

7402 

3789 

58 

^3 

646i 

5o42 

3357 

I4O2 

9176 

6676 

3900 

o846 

75io 

3893 

57 

4 

6607 

5i83 

3493 

i534 

93o4 

6799 

4oi8 

o959 

7619 

3997 

56 

c 

6752 

5324 

3629 

1666 

943i 

6922 

4i36 

1072 

7728 

4ioi 

55 

6 

6897 

5465 

3766 

1798 

9558 

7044 

4254 

n85 

7836 

4204 

54 

7 

7042 

56o5 

3902 

1929 

9685 

7166 

4372 

1299 

7945 

43o8 

53 

8 

7187 

5746 

4o38 

2061 

9812 

7289 

4490 

1412 

8o53 

44i2 

52 

9 

733i 

5886 

4i74 

2192 

9939 

74n 

4607 

i525 

8161 

45i5 

5i 

10 

867476 

876026 

884309 

892323 

900065 

907533 

914725 

921638 

928270 

9346i9 

5o 

ii 

7621 

6167 

4445 

2455 

0192 

7655 

4842 

1750 

8378 

4722 

49 

12 

7765 

6307 

458i 

2586 

oSig 

7777 

4960 

i863 

8486 

4826 

48 

i3 

7910 

6447 

4717 

2717 

o445 

7899 

5°77 

1976 

8594 

4929 

47 

a 

8o54 

6587 

4852 

2848 

©572 

8021 

5i94 

2088 

8702 

5o32 

46 

i5 

8199 

6727 

4988 

2979 

0698 

8i43 

53n 

22OI 

8810 

5i35 

45 

16 

8343 

6867 

5i23 

3no 

0825 

8265 

5429 

23i3 

8917 

5238 

44 

17 

8487 

7006 

5258 

324i 

095  1 

8387 

5546 

2426 

9O25 

534i 

43 

18 

8632 

7146 

5394 

337i 

1077 

85o8 

5663 

2538 

9i33 

5444 

42 

*9 

8776 

7286 

5529 

35o2 

1203 

863o 

5779 

265o 

9240 

5547 

4i 

20 

868920 

877425 

885664 

893633 

901329 

908751 

915896 

922762 

929348 

93565o 

4o 

21 

9064 

7565 

5799 

3763 

i455 

8872 

6oi3 

2875 

9455 

5752 

39 

22 

9207 

77°4 

5934 

3894 

i58i 

8994 

6i3o 

2986 

9562 

5855 

38 

23 

935  1 

7844 

6069 

4024 

1707 

9ii5 

6246 

3o98 

9669 

5957 

37 

24 

9495 

7983 

6204 

4i54 

i833 

9236 

6363 

3210 

9776 

6060 

36 

25 

9639 

8122 

6338 

4284 

i958 

9357 

6479 

3322 

9884 

6162 

35 

26 

9782 

8261 

6473 

44i5 

2084 

9478 

6595 

3434 

999° 

6264 

34 

27 

9926 

84oo 

6608 

4545 

2209 

9599 

6712 

3545 

93oo97 

6366 

33 

28 

870069 

8539 

6742 

4675 

2335 

9720 

6828 

36571   0204 

6468 

3a 

29 

O2  I  2 

8678 

6876 

48o5 

2460 

984i 

6944 

3768 

o3n 

6570 

3i 

3o 

870356 

878817 

887011 

894934 

902585 

9o996i 

917060 

923880 

93o4i8 

936672 

3o 

3i 

0499 

8956 

?i45 

5o64 

2710 

9.10082 

7176 

399i 

o524 

6774 

29 

32 

0642 

9095 

7279 

5i94 

2836 

O2O2 

7292 

4lO2 

o63i 

6876 

28 

33 

o785 

9233 

74i3 

5323 

2961 

0323 

74o8 

42i3 

0787 

6977 

27 

34 

0928 

9372 

7^48 

5453 

3o86 

o443 

7523 

4324 

oS43 

7079 

26 

35 

1071 

95io 

7681 

5582 

3210 

o563 

7639 

4435 

o95o 

7181 

25 

36 

1214 

9649 

78i5 

5712 

3335 

o684 

7755 

4546 

io56 

7282 

24 

37 

1357 

9787 

7949 

584i 

346o 

o8o4 

7870 

4657 

1162 

7383 

23 

38 

1499 

9925 

8o83 

5970 

3585 

0924 

7986 

4768 

1268 

7485 

22 

39 

1642 

88oo63 

8217 

6099 

3709 

1044 

8101 

4878 

i374 

7586 

21 

4o 

871784 

880201 

88835o 

896229 

9<>3834 

911164 

918216 

924989 

93i48o 

937687 

20 

4i 

1927 

o339 

8484 

6358 

3958 

1284 

833i 

5o99 

i586 

7788 

I9 

42 

2069 

0477 

8617 

6486 

4o83 

i4o3 

8446 

5210 

i69i 

7889 

18 

43 

2212 

o6i5 

8751 

66i5 

4207 

i523 

856i 

5320 

i797 

799° 

!7 

44 

2354 

o753 

8884 

6744 

433i 

i643 

8676 

543o 

I902 

8091 

16 

45 

2496 

0891 

9017 

6873 

4455 

1762 

8791 

554i 

2OO8 

8191 

16 

46 

2638 

1028 

9i5o 

7001 

4579 

1881 

8906 

565i 

2Il3 

8292 

i4 

47 

2780 

1166 

9283 

7i3o 

4?o3 

2OOI 

9021 

576i 

22I9 

8393 

i3 

48 

2922 

i3o3 

9416 

?258 

4827 

2I2O 

9i35 

5871 

2324 

8493 

12 

49 

3o64 

i44i 

9549 

7387 

495i 

2239 

9250 

5980 

2429 

8593 

II 

5o 

873206 

881578 

889682 

897515 

co5o75 

912358 

919364 

926090 

932534 

938694 

IO 

01 

3347 

1716 

98i5 

7643 

5i98 

2477 

9479 

6200 

2639 

8794 

9 

52 

3489 

i853 

9948 

777i 

5322 

2596 

9593 

63io 

2744 

8894 

8  < 

53 

363i 

1990 

890080 

•7900 

5445 

2715 

9707 

64i9 

2849 

8994 

7 

54 

377a 

2127 

02l3 

8028 

5569 

2834 

9821 

6529 

2954 

9094 

6 

55 

39i4 

2264 

o345 

8i56 

5692 

2953 

9936 

6638 

3o58 

9i94 

5 

56 

4o55 

2401 

0478 

8283 

58i5 

3072 

92oo5o 

6747 

3i63 

9294 

4 

57 

4196 

2538 

0610 

84n 

5939 

3190 

oi64 

6857 

3267 

9394 

3 

58 

4338 

2674 

0742 

8539 

6062 

3309 

0277 

6966 

3372 

9493 

2 

59 

4479 

2811 

0874 

8666 

6i85 

3427 

o39i 

7o75 

3476 

9593 

1 

29° 

28°   27° 

26° 

25° 

24°   23° 

22°   21° 

20° 

a 

Natural  Co-sines. 

§ 

jfe 

2.3l 

2.24 

2.  16 

2.09 

2.OI 

i.93 

1.86 

1.78 

1.70 

NATURAL    TANGENTS. 


d 
1 

20° 

61° 

62° 

63° 

64° 

65° 

66° 

67°   68° 

69-  > 

o 

1.73205 

i.8o4o5 

1.88073 

1.96261 

2.o5o3o 

2.i445i 

2.  246o4 

2.35585 

2.47509 

2.6o5o9|6o 

I 

332i 

0529 

82o5 

6402 

5i82 

46i4 

4780 

5776 

7716 

0736I59 

2 

3438 

o653 

8337 

6544 

5333 

4777 

4g56 

5967 

7924 

o963 

58 

3 

3555 

°777 

8469 

6685 

5485 

4940 

5i32 

6i58 

8i32 

1190 

57 

4 

367i 

0901 

8602 

682-7 

5637 

5io4 

5309 

6349 

834o 

i4i8 

56 

5 

3788 

ios5 

8734 

6969 

579o 

5268 

5486 

654i 

8549 

1  646 

55 

6 

3905 

n5o 

8867 

7in 

5942 

5433 

5663 

6733 

8758 

187^ 

54 

7 

4022 

1274 

9000 

7253 

609^ 

5596 

584o 

6925 

8967 

2IO3 

53 

8 

4i4o 

1  399 

9i33 

7395 

6247 

576o 

6018 

7n8 

9177 

2332 

52 

9 

4257 

i524 

9266 

7538 

64oo 

5925 

6196 

73n 

9386 

a56i 

5i 

10 

1.74375 

1.81649 

1.89400 

1.97681 

2.o6553 

2.16090 

2.26374 

2.375o4 

2.49597 

2.62791 

5o 

ii 

4492 

1774 

9533 

7823 

67o6 

6255 

6552 

7697 

9807 

3021 

49 

12 

46io 

1899 

9667 

7966 

6860 

6420 

673o 

7891 

a.5ooi8 

3252 

48 

i3 

4728 

2O25 

9801 

8110 

7oi4 

6585 

6909 

8o84 

0229 

3483 

47 

:4 

4846 

2i5o 

9935 

8253 

7167 

675i 

7088 

8279 

o44o 

3714 

46 

i5 

4964 

2276 

1.90069 

8396 

7321 

6917 

7267 

8473 

o652 

3945 

45 

16 

5o8a 

24O2 

O2O3 

854o 

7476 

7o83 

7447 

8668 

o864 

4177 

44 

17 

5200 

2528 

o337 

8684 

763o 

7249 

7626 

8863 

io76 

44io 

43 

18 

5319 

2654 

0472 

8828 

7785 

7416 

7806 

9o58 

1289 

4642 

4a 

J9 

5437 

2780 

0607 

8972 

7939 

7582 

7987 

9253 

i5o2 

4875 

4i 

20 

i.75556 

1.82906 

1.90741 

i  .99  1  1  6 

2.08094 

2.17749 

2.28167 

2.39449 

2.5i7i5 

2.65io9 

4o 

21 

5675 

3o33 

0876 

9261 

825o 

7916 

8348 

9645 

1929 

5342 

39 

22 

5794 

3i59 

1012 

9406 

84o5 

8084 

8528 

9841 

2l42 

5576 

38 

23 

5,913 

3286 

1147 

955o 

856o 

825i 

8710 

2.4oo38 

2357 

58n 

37 

24 

3o32 

34i3 

1282 

9695 

8716 

8419 

8891 

0235 

2571 

6o46 

36 

25 

6i5i 

354o 

i4i8 

9841 

8872 

8587 

9o73 

0432 

2786 

6281 

35 

26 

6271 

3667 

i554 

9986 

9028 

8755 

9254 

0629 

3ooi 

65i6 

34 

27 

GSgo 

3794 

1690 

2.OOl3l 

9184 

8923 

9437 

0827 

3217 

6752 

33 

28 

65io 

3922 

1826 

0277 

934i 

9092 

9619 

IO25 

3432 

6989 

32 

29 

663o 

4049 

1962 

0423 

9498 

9261 

9801 

1223 

3648 

7225 

3i 

3o 

1.76749 

1.84177 

1.92098 

2.00669 

2.09654 

2.19430 

2.29984 

2.41421 

2.53865 

2.67462 

3o 

3i 

6869 

43o5 

2235 

o7i5 

9811 

9599 

2.30167 

l62O 

4082 

7700 

29 

32 

6990 

4433 

2371 

0862 

9969 

9769 

o35i 

1819 

4299 

7937 

28 

33 

7110 

456i 

25o8 

1008 

2.10126 

9938 

o534 

2OI9 

45i6 

8  -.-5 

27 

34 

723o 

4689 

2645 

ii55 

0284 

2.20108 

0718 

22l8 

4734 

84i4 

26 

35 

735i 

48i8 

2782 

1302 

0442 

0278 

0902 

2418 

4g52 

8653 

25 

36 

7471 

4946 

2920 

1  449 

0600 

o449 

1086 

2618 

6170 

8892 

24 

37 

7592 

5o75 

3o57 

1596 

o758 

0619 

1271 

2819 

5389 

9i3i 

23 

38 

77i3 

52o4 

3ig5 

1743 

0916 

0790 

i456 

3019 

56o8 

937i 

22 

39 

7834 

5333 

3332 

1891 

1075 

0961 

i64i 

3220 

5827 

9612 

21 

4o 

i.  77955 

1.85462 

1.93470 

2.02039 

2.II233 

3.2II32 

2.31826 

2.43422 

2.56o46 

2.69853 

20 

4i 

8077 

559i 

36o8 

2187 

1392 

i3o4 

2OI2 

3623 

6266 

2.70094 

*9 

4s 

8198 

5720 

3746 

2335 

155? 

i475 

2I97 

3825 

6487 

o335 

18 

43 

8319 

585o 

3885 

2483 

1711 

1  647 

2383 

4027 

6707 

o577 

ll 

44 

844i 

5979 

4023 

263i 

1871 

1819 

2570 

4230 

6928 

0819 

16 

45 

8563 

6109 

4162 

2780 

2o3o 

1992 

2756 

4433 

7i5o 

3062 

i5 

46 

8685 

6239 

43oi 

2929 

2190 

2i64 

2943 

4636 

737i 

i3o5 

i4 

4? 

8807 

6369 

444o 

3078 

235o 

2337 

3i3o 

4839 

7593 

1  548 

i3 

48 

8929 

6499 

4579 

3227 

25ll 

25lO 

33i7 

5o43 

78i5 

I792 

12 

49 

9o5i 

663o 

47i8 

3376 

2671 

2683 

35o5 

5246 

8o38 

2o36 

II 

5o 

1.79174 

1.86760 

1-94858 

2.o3526 

2.12832 

2.22857 

2.33693 

2.4545i 

2.58261 

2.72281 

10 

5i 

9296 

6891 

4997 

3675 

2993 

3o3o 

388i 

5655 

8484 

2526 

9 

52 

9419 

7021 

5i37 

3825 

3i54 

32o4 

4069 

586o 

8708 

2771 

8 

53 

9542 

7i52 

6277 

3975 

33i6 

3378 

4258 

6o65 

8932 

3017 

7 

54 

9665 

7283 

54i7 

4i25 

3477 

3553 

4447 

6270 

9i56 

3263 

6 

55 

9788 

74i5 

5557 

4276 

3639 

3727 

4636 

6476 

938i 

3509 

5 

56 

9911 

7546 

5698 

4426 

38oi 

3902 

4825 

6682 

96o6 

3756 

4 

57 

i.8oo34 

7677 

5838 

4577 

3g63 

4077 

5oi5 

6888 

983! 

4oo4 

3 

58 

oi58 

7809 

5979 

4728 

4i25 

4252 

52o5 

7o95 

2.  6oo57 

425i 

2 

~ 

0281 

794  1 

6120 

4879 

4288 

4428 

5395 

7302 

0283 

4499 

I 

29° 

28° 

27° 

26° 

25° 

24° 

23° 

22° 

21° 

20° 

S' 

Natural  Co-tangents. 

s 

P.  P 

to  I".2'00 

2.l3 

2.27 

2.44 

2.62  !  2.82 

3.o5 

3.3i 

3.6i 

3.95 

' 

t  30 


NATURAL    SINKS. 


e 

1 

70° 

71° 

72° 

73° 

74° 

75° 

7(5° 

77° 

78° 

79° 

o 

939693 

9455i9 

95io57 

9563o5 

961262 

965926 

970296 

97437o 

978148 

081627 

60 

I 

9792 

56i3 

n46 

6390 

1  342 

6001 

o366 

4435 

8208 

i683 

59 

2 

9891 

57o8 

1236 

6475 

l422 

6o76 

o436 

45oi 

8268 

i738 

58 

3 

9991 

58o2 

i326 

656o 

1502 

6i5i 

o5o6 

4566 

8329 

I793 

57 

4 

940090 

5897 

i4i5 

6644 

i582 

6226 

o577 

463i 

8389 

i849 

56 

5 

0189 

599i 

i5o5   6729 

1662 

63oi 

0647 

4696 

8449 

i9o4 

55 

6 

0288 

6o85 

I5941   6814 

i74i 

6376 

0716 

476i 

8509 

i959 

54 

7 

o387 

6180 

1684;   6898 

1821 

645  1 

0786 

4826 

8569 

2Ol4 

53 

8 

o486 

6274 

i773 

6983 

1901 

6526 

o856 

489i 

8629 

2o69 

52 

9 

o585 

6368 

1862 

7o67 

1980 

6600 

0926 

4956 

8689 

2123 

5i 

10 

g4o684 

946462 

951951 

957i5i 

962059 

966675 

97°995 

975o2o 

978748 

982i78 

5o 

n 

0782 

6555 

2O4O 

7235 

2139 

6749 

io65 

5o85 

8808 

2233 

49 

12 

0881 

6649 

2129 

73i9 

2218 

6823 

u34 

5i49 

88$7 

2287 

48 

i3 

0979 

6743 

2218 

74o4 

2297 

6898 

1204 

52i4 

8qi- 

3342 

47 

i4 

1078 

6837 

2307 

7487 

2376 

6972 

1273 

5278 

8986 

2396 

46 

i5 

1176 

6930 

2396;  7571 

2455 

7046 

I  342 

5342 

9045 

245o 

45 

16 

1274 

7024 

2484!   7655 

2534 

7120 

i4n 

54o6 

9io5 

25o5 

44 

i? 

1372 

7117 

2573 

7739 

26i3 

7194 

1480 

547i 

9164 

2559 

43 

18 

1471 

7210 

2661 

7822 

2692 

7268 

1  549 

5535 

9223 

26i3 

42 

is! 

i569 

73o4 

2750 

7906 

2770 

7342 

1618 

5598 

9282 

2667 

4i 

20 

941666 

947397 

952838 

957990 

962849 

967415 

971687 

975662 

979341 

982-721 

4o 

21 

1764 

7490 

2926 

8o73 

2928 

7489 

i755 

5726 

9399 

2774 

39 

22 

1862 

7583 

3oi5 

8i56 

3oo6 

7562 

1824 

579o 

9458 

2828 

38 

23 

1960 

7676 

3io3 

8239 

3o84 

7636 

i893 

5853 

95l7 

2882 

37 

24 

2057 

7768 

3191!   8323 

3i63 

7709 

1961 

59i7 

9575 

2935 

36 

25 

2i55 

7861 

3279 

84o6 

324l 

7782 

2029 

5980 

9634 

2989 

35 

26 

2252 

7954 

3366 

8489 

33i9 

7856 

2098 

6o44 

9692 

3o42 

34 

27 

235o 

8o46 

3454 

8572 

3397 

7929 

2166 

6io7 

975o 

3096 

33 

28 

2447 

8i39 

3542 

8654 

3475 

8002 

2234 

6i7o 

98o9 

3i49 

32 

29 

2544 

823i 

3629 

8737 

3553 

8o75 

2302 

6233 

9867 

3eo2 

3i 

3o 

942641 

948324 

9537i7 

958820 

96363o 

968148 

972370 

976296 

979925 

983255 

3o 

3i 

2739 

84i6 

38o4 

8902 

37o8 

8220 

2438 

6359 

9983 

33o8 

29 

32 

2836 

85o8 

3892   8g85 

3786 

8293 

25o6 

6422 

98oo4i 

336i 

28 

33 

2932 

8600 

3979 

9o67 

3863 

8366 

2573 

6485 

oo98 

34i4 

27 

34 

3029 

8692 

4o66 

9i5o 

394i 

8438 

2641 

6547 

oi56 

3466 

26 

35 

3i26 

8784 

4i53 

9232 

4oi8 

85n 

2708 

6610 

02l4 

35i9 

25 

36 

3223 

8876 

424o 

93i4 

4o95 

8583 

2776 

6672 

027I 

357i 

24 

37 

33i9 

8968 

4327 

9396 

4i73 

8656 

2843 

6735 

o329 

3624 

23 

38i 

34i6 

9059 

44i4 

9478 

4260 

8728 

2911 

6797 

o386 

3676 

22 

39 

35ia 

9i5i 

45oi 

956o 

4327 

8800 

2978 

6859 

o443 

3729 

21 

4o 

943609 

949243 

954588 

959642 

9644o4 

968872 

973o45 

97692i 

gSoSoo 

983781 

2O 

4i 

37o5 

9334 

4674 

9724 

448  1 

8944 

3lI2 

6984 

o558 

3833 

'9 

42 

38oi 

9425 

4761 

9805 

4557 

*  9016 

3i79 

7o46 

o6i5 

3885 

18 

43 

3897 

95l7 

4847 

988-7 

4634 

9088 

3246 

7108 

o672 

3937 

17 

44 

3993 

9608 

4934 

9968 

47n 

9i59 

33i3 

7169 

0728 

3989 

16 

45 

4o89 

9699 

502O 

96oo5o 

4787 

9231 

3379 

7231 

o785 

4o4i 

i5 

46 

4i85 

979° 

5io6 

oi3i 

4864 

9302 

3446 

7293 

0842 

4092 

i4 

47 

4281 

-9881 

5192 

O2I2 

494o 

9374 

35i2 

7354 

0899 

4i44 

i3 

43 

4376 

9972 

5278 

O294 

5oi6 

9445 

3579 

74i6 

ogSS 

4196 

12 

49 

4473 

95oo63 

5364 

o375 

5o93 

95i7 

3645 

7477 

1012 

4247 

II 

•  5o 

944568 

95oi54 

95545o 

960456 

965169 

969588 

9737i2 

977539 

981068 

984298 

10 

5i 

4663 

0244 

5536 

o537 

5245 

9659 

3778 

7600 

1124 

435o 

9 

52 

4758 

o335 

5622 

0618 

532i 

973o 

3844 

7661 

1181 

44oi 

8 

53 

4854 

o4a5 

5707 

0698 

5397 

9801 

39io 

7722 

I237 

4452 

7 

54 

4949 

o5i6 

5793 

°779 

5473 

9872 

3976 

7783 

1293 

45o3 

6 

55 

5o44 

0606 

5879 

0860 

5548   9943 

4042 

7844 

i349 

4554 

5 

56 

5i39 

0696 

5964 

og4o 

5624  97ooi4 

4io8 

7906 

i4o5 

46o5 

4 

57 

5234 

0786 

6o49 

1021 

57oo   oo84 

4i73 

7966 

i46o 

4656 

3 

58 

5329 

0877 

6i34 

IIOI 

5775!   oi55 

4239 

8026 

i5i6 

4707 

2 

59 

5424 

0967 

6220 

1181 

585o   0225 

43o5 

8087 

l572 

4757 

I 

19° 

18° 

17° 

16°    15°    14° 

13° 

12° 

11°   10° 

d 

Natural  Co-sines. 

S 

£i->62 

i.54   i.46 

i.38 

:.3o 

I.  21 

i.i3 

i.o5 

o.97 

0.88 

N  A  T  URAL     T  ,\       GENTS. 


13! 


d 

70° 

71° 

72° 

73° 

74° 

75° 

76° 

77° 

78° 

79° 

o 

2.74748 

2.9042I 

3.07768 

3.27o85 

3.4874i 

3.73205 

4.01078 

4.33:48 

4-70463 

5.i4455 

60 

I 

4997 

o696 

8073 

7426 

9I25 

364o 

i576 

3723 

n37 

5256 

59 

2 

5246 

0971 

8379 

7767 

95o9 

4o75 

2074 

43oo 

i8i3i   6o58 

58 

3 

5496 

1246 

8685 

8109 

9894 

45i2 

2574 

4879 

2490 

6863 

57 

4 

5746 

i523 

8991 

8452 

3.5o279 

495a   3o76 

5459!   3i7o 

767i 

56 

5 

5996 

1799 

9298 

8795 

o666|   5388|   3578 

6o4o 

385i 

848o 

55 

6 

6247 

2076 

96o6 

9i39 

io53 

5828 

4o8i 

6623 

4534 

9293 

54 

7 

6498 

2354 

9914'  9483;   i44i 

6268 

4586 

7207 

52I9 

5.20107 

53 

8 

675o 

2632 

3.IO223    9829 

1829 

67o9 

5o92 

7793 

59o6 

0925 

52 

9 

7002 

2910 

o532 

3.3oi74 

2219 

7l52 

5599 

838i 

6595 

1744 

5i 

10 

2.77254 

2.93189 

3.io842 

3.3o52i 

3  52609 

3.77595 

4.06107 

4.38969 

4.77286 

5.22566 

5o 

n 

75°7 

3468 

n53 

0868 

3ooi 

8o4o 

6616 

956o 

7978 

339i 

49 

12 

7761 

3748 

i464 

1216 

3393 

8485 

7I27 

4-4oi52 

8673 

4218 

48 

i3 

8oi4 

4028 

i775 

i565 

3785 

893i 

7639 

o745 

937o 

5o48 

47 

i4 

8269 

43o9 

2087 

1914 

4i79 

9378 

8i52 

i34o 

4.80068 

588o 

46 

i5 

8523 

459i 

2400 

2264 

4573 

9827 

8666 

i936 

o769 

67i5 

45 

16 

•  8778 

4872 

27131   2614 

4968 

3.8o276 

9l82 

2534 

i47i 

7553 

44 

J7 

9o33 

5i55 

3027 

2965 

5364 

O726 

9699 

3i34 

2I75 

8393 

43 

18 

9289 

5437 

334i 

33i7 

576i 

II77 

4.10216 

3735 

2882 

9235 

42 

J9 

9545 

5721 

3656 

367o 

6i59 

i63oj   o736 

4338 

359o 

5.3oo8o 

4i 

20 

2.79802 

2.96004 

3.13972 

3.34023 

3.56557 

3.82083 

4-  1  1256  4.44942 

4.843oo 

5.3o928 

4o 

21 

2.80059 

6288 

4288 

4377 

6957 

2537 

1778 

5548 

5oi3 

1778 

39 

22 

o3i6 

6573 

46o5 

4732 

7357 

2992 

2301 

6i55 

5727 

263i 

38 

23 

o574 

6858 

4922 

5o87 

7758 

3449 

2825 

6764 

6444 

3487 

37 

24 

o833|   7i44 

524o 

5443 

8160 

39o6 

335o 

7374 

7l62 

4345 

36 

25 

1091 

743o 

5558 

58oo 

8562 

4364 

3877 

7986 

-7882 

5206 

35 

20 

i35o 

7717 

5877 

6i58 

8966 

4824 

44o5 

8600 

86o5 

6070 

34 

27 

1610 

8oo4 

6197,   65i6i   9370 

5284 

4934 

92l5 

933o 

6936 

33 

2« 

1870 

8292 

65i7 

6875i   9775 

5745 

5465 

9832 

4-9oo56 

78o5 

32 

29 

2i3o 

858o 

6838 

7234'3.  60181 

6208 

5997 

4.5o45i 

o785 

8677 

3i 

3o 

2.82391 

2.98868 

3.i7i59 

3.375943.6o588 

3.8667i 

4.i653o 

4.5io7i 

4-9i5i6 

5.39552 

3o 

3i 

2653 

9i58i   7481 

7955;   o996 

7i36 

7064 

i693 

2249 

5.4o429 

29 

32 

2914 

9447'   7804 

83i7|   i4o5 

76oi 

7600 

23i6 

2984 

i3o9 

28 

33 

3i76 

9738 

8127   8679i   1814 

8068 

8137 

294i 

372I 

2I92 

27 

34 

3439 

3.00028 

845  1  1   9o42 

2224 

8536 

8675 

3568 

446o 

3o77 

26 

35 

3702 

0319 

8775|   94o6 

2636 

9oo4 

92l5 

4196 

52OI 

3966 

25 

36 

3965 

0611 

9ioo 

9771 

3o48 

9474 

9756 

4826 

5945 

4857 

24 

37 

4229 

0903 

9426 

3.4oi36 

346  1 

9945 

4.20298 

5458 

669o 

575i 

23 

38 

4494 

1196 

9752 

o5o2 

3874 

3.9o4i7 

0842 

6091 

7438 

6648 

22 

39 

4758 

i489 

3.2OO79 

0869 

4289 

o89o 

i387 

6726 

8188 

7548 

21 

4o 

2.85023 

3.oi783 

3.2o4o6 

3.4i236 

3.647o5 

3.9i364 

4-2i933 

4.57363 

4.9894o 

5.4845i 

2O 

4i 

5289 

2077 

o734 

1604 

5l2I 

i839 

2481 

8001 

9695 

9356 

I9 

42 

5555 

2372 

io63 

i973 

5538 

23i6 

3o3o 

864i 

5.oo45i 

5.50264 

18 

43 

5822 

2667 

l392 

2343 

5957 

2793 

358o'   9283   1210 

1176 

!? 

44 

6089 

2963 

1722 

27l3 

6376 

3271 

4l32 

9927 

i97i 

2090 

16 

45 

6356 

3260 

2o53 

3o84 

6796 

375i 

4685 

4«6o572 

2734 

3007 

i5 

46 

6624 

3556 

2384 

3456 

72I7 

4232 

5239 

1219 

3499 

3927 

i4 

47 

6892 

3854 

2715 

3829 

7638 

4713 

5795 

1868 

4267 

485i 

i3 

48 

7161 

4i52 

3o48 

4202 

8061 

5i96 

6352 

25i8 

5o37 

5777 

12_ 

49 

743o 

445o 

338i 

4576 

8485 

568o 

69n 

3i7i 

58o9 

6706 

II 

5o 

2.87700 

3.04749 

3.23714 

3.4495i 

3.689o9 

3.96i65 

4.27471 

4.63825 

5.o6584 

5  57638 

IO 

5i 

7970 

5o49 

4o49 

5327 

9335 

665i 

8o32 

448o 

736o 

8573 

9 

52 

8240 

5349 

4383 

57o3 

976i 

7i39 

8595 

5i38 

8139 

95n 

8 

53 

85n 

5649 

4719'   6080 

3.7oi88 

7627 

9i59 

5797 

892I 

5.6o452 

7 

54 

8783 

595o 

5o55   6458 

0616 

8117 

9724 

6458   97o4 

i397 

6 

55 

9o55 

6252 

5392 

6837 

io46 

8607 

4.3o29i   7i2i  5.io49o   a344 

5 

56 

9327 

6554 

5729 

•7216 

i476 

9o99   0860'   7786   i279   3a95 

4 

57 

9600 

6857 

6067 

7596 

I9°7 

9592   i43o;   8452   2o69   4248 

3 

58 

9873 

7160 

64o6 

7977 

2338  4.00086   2001   9i2i   2862   02o5 

2 

59 

2.90147 

7464 

6745 

8359 

2771   o582   2573   979i   3658   6i65 

I 

19° 

18° 

17° 

16° 

15°    14°    13°    12°    11°    10° 

9' 

Natural  Co-tangents. 

S 

££4.35 

4.82 

5.36 

6.01 

6.79 

7.73 

8.9o 

10.35 

12.20 

i4.6o 

132 


NATURAL    SINES. 


a 

80° 

81°  ,  S2° 

S30 

84° 

85° 

86° 

87° 

qg°   89° 

0 

9848o8 

987686 

990268)  992546 

994522 

996195 

997564 

998680 

999891 

999848 

60 

I 

4858 

7734 

o3o9 

2582 

4552 

6220 

7584 

8645 

9401 

9853 

5o 

2 

49o9 

7779 

o34g 

26l7 

4583 

6245 

7604 

8660 

9411 

9358 

58 

3 

4959 

7824 

o389 

2602 

46i3 

62-70 

7625 

8675 

9421 

9863 

57 

4 

5oo9 

7870 

04291   268-7 

4643 

6295 

7645 

8690 

943i 

9867 

56 

5 

5o59 

79i5 

0469 

2722 

4673 

6320 

7664 

87o5 

944  1 

9872 

55 

6 

5io9 

7960 

oSog 

2757 

4703 

6345 

7684;  8719 

945o 

9877 

54 

7 

5i59 

8oo5 

0549 

2792 

4733 

637o 

77°4 

8734 

9460 

9881 

53 

8 

5209 

8o5o 

0589 

2827 

4762 

6395 

7724 

8749 

9469 

9886 

52 

9 

5259 

8094 

0629 

2862 

4792 

6419 

7743 

8763 

9479 

9890 

5i 

10 

9853o9 

988139 

990669 

992896 

994822 

996444 

997763 

998778 

999488 

999894 

5o 

;i 

5358 

8184 

o7o8 

293l 

485  1 

6468 

7782 

8792 

9497 

9898 

49 

12 

54o8 

8228 

o748 

2966 

488i 

6493 

7801 

8806 

9507 

9903 

48 

13 

5457 

8273 

o787 

3ooo 

4910 

65i7 

7821 

8820 

95i6 

9907 

47 

i4 

55o7 

83i7 

o8a7 

3o34 

4939 

654! 

7840 

8834 

9525 

9910 

46 

i5 

5556 

8362 

0866 

3o68 

4969 

6566 

7859 

8848 

9534 

9914 

45 

16 

56o5 

84o6 

ogoS 

3io3 

4998 

6590 

7878 

8862 

9542 

9918 

44 

17 

5654 

845o 

0944 

3i37 

5027 

66i4 

7897 

8876 

955i 

9922 

43 

18 

57o3 

8494 

0983 

3i7i 

5o56 

6637 

79i6 

889o 

956o 

9925 

42 

19 

575a 

8538 

IO22 

32o5 

5o84 

6661 

7934 

89o4 

9568 

9929 

4i 

20 

9858oi 

988582 

991061 

993238 

995n3 

996685 

997953 

098917 

999577 

999932 

4o 

21 

585o 

8626   noo 

3272 

5i42 

6709 

7972 

8931 

9585 

9936 

39 

22 

5899 

8669,   n38;   33o6 

5170 

6732 

799° 

8944 

9594 

9939 

38 

23 

5947 

87131   1177   3339 

5199 

6756 

8008 

8957 

96o2 

9942 

37 

24 

5996 

8756i   1216'   3373 

5227 

6779 

8027 

8971 

96io 

9945 

36 

25 

6o45 

8800 

1254 

34o6 

5256 

6802 

8o45 

8984 

96i8 

9948 

35 

26 

6o93 

8843 

1292 

3439 

5284 

6825 

8o63 

8997 

9626 

995i 

34 

27 

6i4i 

8886 

i33i 

3473 

53i2 

6848 

8081 

9010 

9634 

9954 

33 

28 

6i89 

893o 

i36g 

35o6 

534o 

6872 

8o99 

9023 

9642 

9957 

32 

29 

6238 

8973 

i4o7 

3539 

5368 

6894 

8117 

9o35 

965o 

9959 

3i 

3o 

986286 

989016 

99i445 

993572 

995396 

996917 

998i35 

999048 

999657 

99996s 

3o 

3i 

6334 

9059 

i483 

36o5 

5424 

6940 

8i53 

9061 

9665 

9964 

29 

32 

638i 

9102 

l52I 

3638 

5452 

6963 

8170 

9o73 

9672 

9967 

28 

33 

6429 

9145 

i558 

367o 

5479 

6985 

8188 

9086 

968o 

9969 

27 

34 

6477 

9187   i596 

37o3 

55o7 

7008 

8205 

9098 

9687 

9971 

26 

35 

6525 

923o*   i634 

3735 

5535 

7o3o 

8223 

9111 

9694 

9974 

25 

36 

6572 

9272   1671 

3768 

5562 

7o52 

8240 

9123 

9701 

9976 

24 

37 

6620 

93i5   17°9 

38oo 

5589 

7°75 

8257 

9I35 

9709 

9978 

23 

38 

6667 

9357   i746 

3833 

56i7 

7097 

8274 

9716 

998o 

22 

39 

67:4 

9399   i783 

3865 

5644 

7n9 

829I 

9l59 

9722 

998i 

21 

4o 

991820 

993897 

99567i 

997i4i 

9983o8 

9991?1 

999729 

9V9983 

2O 

4i 

68o9 

9  9484 

i857 

3929 

5698 

7163 

8325 

9i83 

9736 

9985 

19 

42 

6856 

9526 

1894 

396i 

5725 

7i85 

8342 

9194 

9743 

9986 

18 

43 

69o3 

9568 

i93i 

3993 

5752 

7207 

8359 

9206 

9749 

9988 

17 

44 

695o 

96io 

1968 

5778 

7229 

8375 

9218 

9756 

9989 

16 

45 

6996 

965i 

2OO5 

4o56 

58o5 

725o 

8392 

9229 

9762 

9990 

i5 

46 

9693 

2042 

4o88 

5832 

7272 

84o8 

9240 

9768 

9992 

i4 

47 

7o9o 

9735 

2078 

4l2O 

5858 

7293 

8425 

9252 

9775 

9993 

i3 

48 
49 

7i36 
7i83 

97761   2ii5 
•9818!  2i5i 

4.i5i 
4182 

5884 
Sgii 

73i4 
7336 

844i 
8457 

9263 

9274 

9781 
9787 

9994 
9995 

12 
II 

5o 

989859 

992187 

994214 

995937 

997357 

998473 

999285 

999793 

979996 

10 

5i 

7275 

9900 

2224 

4245 

5963 

7378 

8489 

9296 

9799 

9997 

9 

52 

7322 

9942 

2260 

4276 

5989 

7399 

85o5 

9307 

980^ 

9997 

8 

53 

7368 

9983 

2296 

43o7 

6oi5 

7420 

852i 

93i8 

98io 

9998 

7 

54 

74i4 

99002^ 

2332 

4338 

6o4i 

744i 

8537|   9328 

98i6 

9998 

6 

55 

746o 

oo65 

2368 

4369 

6o67 

7462 

855a 

9339 

982I 

9999 

5 

56 

75o6 

oio5 

240^ 

44oo 

6o93 

7482 

8568 

935o 

9827 

99991  4 

57 

755i 

oi46 

243g 

443o 

6118 

75o3 

8583 

936o 

9832 

I.OOOOO 

3 

58 

7597 

0187 

247£ 

446  1 

6i4-4 

752^ 

8599 

937o 

9837 

0000 

2 

59 

7643 

0228 

25ll 

449i 

6i6c, 

7544 

86i4 

938i 

9843 

ooooj  i 

9° 

8° 

7° 

6° 

5° 

.  4° 

3° 

2° 

1° 

°°  !  d 

Natural  Co-sines. 

i 

P.  P  o  go 

0.72 

o.63 

o.55 

0.46 

o.38 

o.3o 

0.21 

o.i3 

o.o4 

' 

J 

NATURAL    TANGENTS. 


i  33 


i 

80° 

81° 

82W 

83° 

84° 

85° 

86° 

87° 

88° 

89°  | 

0 

5.671286.31376 

7.11537 

8.i4435 

9.5i436 

n.43oi 

14.3007 

19.0811 

28.6363 

67.2900 

txO 

I 

8o94;   2666)   3o42 

6398 

4io6 

4685 

3607 

1879 

8771 

68.2612 

59 

2 

9o64<   376i 

4553 

837o 

6791 

6072 

4212 

2969 

29.1220 

69.2669 

58 

3 

6.70037!   496i 

6071 

8.20352 

9490 

546i 

4823 

4o5i 

3711 

6o.3o58 

57 

4 

ioi3(   6i65 

7694 

2344 

9.  62206 

5853 

5438 

6166 

6246 

61.3829 

56 

5 

1992 

7374 

9126 

43451   4935 

6248 

6069 

6273 

8823 

62.4992 

55 

6 

2974 

8587 

7.20661 

6355 

7680!   6645 

6685 

74o3 

3o.i446 

63.6667 

54 

7 

3960   g8o4 

22O4 

8376 

9.7o44i   7046 

73i7 

8546 

4n6 

64.8680 

53 

8 

4949 

6.41026 

3754 

8.3o4o6 

3217;  7448 

7964   9702 

6833 

66.io55 

62 

9 

594i 

2253 

53io 

2446 

6oo9 

7853 

8696 

20.0872 

9699 

67.4019 

5i 

10 

5.76937 

6.43434 

7.26873 

8.34496 

9.788i7 

11.8262 

14.9244 

20.2066 

68.7601 

5o 

ii 

7936 

4720 

8442 

6555 

9.81641 

8673 

9898 

3253 

6284 

70.i533 

49 

12 

8938 

596i 

7.30018 

8626 

4482 

9087 

i5.o557 

4465 

8206 

71.6161 

48 

i3 

9944 

"7206 

1600 

8.4o7o5 

7338 

1222 

6691 

32.n8i 

73.1390 

47 

i4 

5.80963 

8456 

3190 

2796 

9.90211 

9923 

i8q3 

6932 

4213 

74.7292 

46 

i5 

1966 

97io 

4786 

4896 

3ioi 

I2.o346 

2671 

8188 

73o3 

76.3900 

45 

16 

2982 

6.5o97o 

6389 

7007 

6007 

0772 

3254 

946o 

33.0452 

78.1263 

44 

ll 

4ooi 

2234 

7999 

9128 

893i 

I2OI 

3943 

21.0747 

3662 

79.9434 

43 

18 

6024 

35o3 

9616 

8.61269 

10.0187 

i632 

4638 

2049 

6935 

81.8470 

42 

J9 

6061 

4777 

7.41240 

3402 

o483 

2067 

534o 

3369 

34.0273 

83.8435 

4i 

20 

6.87080 

6.56o55 

7.42871 

8.55555 

10.0780 

12.2606 

i5.6o48 

21.4704 

34.3678 

85.9398 

4o 

21 

8n4 

7339 

4609 

7718 

1080 

2946 

6762 

6066 

7161 

88.i436 

39 

22 

9161 

8627 

6i54 

9893 

i38i 

3390 

7483 

7426 

35.o695 

9o.4633 

38 

23 

6.90191 

992I 

7806 

8.62078 

i683 

3838 

8211 

88i3 

43i3 

9,2.9086 

37 

24 

1236 

6.6i2i9 

9465 

4276 

1988 

4288 

8945 

22.0217 

8006 

96.4896 

36 

25 

2283 

2623 

7.5n32 

6482 

2294 

4742 

9687 

1640 

36.i776 

98.2179 

35 

26 

3335 

383i 

2806 

8701 

2602 

6199 

i6.o435 

3o8i 

5627 

101.107 

34 

27 

439o 

5x44 

4487 

8.70931 

2913   6660 

II9O 

454i 

956o 

104.171 

33 

28 

5448 

6463 

6176 

3172 

3224'    6l24 

I952 

6020 

37.3579 

107.426 

32 

29 

65io 

7787 

7872 

6426 

3538   6691 

2722 

7619 

7686 

110.892 

3i 

3o 

5.9757€ 

6.69n6 

7.59676 

8.77689 

10.3864112.7062 

16.3499 

22.9038 

38.1885 

114.689 

3o 

3i 

8646 

6.70460 

7.61287 

9964 

4172 

7536 

4283 

23.0677 

6177 

n8.54o 

29 

32 

9720 

•  1780 

3oo5 

8.82262 

4491 

8oi4 

5o75 

2137 

39.o568 

122.774 

28 

33 

6.00797 

3i33 

4732 

455i 

48i3 

8496 

5874 

3718 

5o59 

127.321 

27 

34 

1878 

4483 

6466 

6862 

5i36 

8981 

6681 

5321 

9655 

132.219 

26 

35 

2962 

5838 

8208 

9186 

6462 

9469 

7496 

6945 

40.4358 

137.607 

25 

36 

4o5i 

7i99 

9967 

8.91620 

6789 

9962 

83i9 

8593 

9174 

143.237 

24 

37 

5i43 

8564 

7.7i7i5 

3867 

6118 

i3.o458 

9i5o 

24.0263 

4i.4io6 

149-465 

23 

38 

6240 

9936 

348o 

6227 

645o 

0968 

999° 

1967 

9168 

166.269 

22 

39 

734o 

6.8i3i2 

6264 

8698 

6783 

i46i 

I7.o837 

3675 

42.4335 

163.700 

21 

4o 

6.o8444 

6.82694 

7.77o35 

q.ooqSS 

10.7119 

13.1969 

1-7.1693 

24.54i8 

42.9641 

171.886 

2O 

4i 

9662 

4082 

8826 

3379 

7467 

2480 

2558J   7i85 

43.5o8i 

180.932 

I9  I 

42 

6.10664 

5475 

7.8o622 

6789 

7797 

2996 

3432!   89-78 

44.o66i 

190.984 

18 

43 

1779 

68  74 

2428 

8211 

8139 

35i5 

43i4'25.o798 

6386 

202.219 

17 

44 

2899 

8278 

4242 

9.10646 

8483 

4039 

6206   2644 

46.2261 

214.868 

16 

45 

4oa3 

9688 

6o64 

SogS 

8829 

4566 

6106 

45i7 

8294 

229.182 

i5 

46 

5i5i 

6.9no4 

7895 

5554 

9178 

6098 

7016 

64i8 

46.4489 

246.662 

i4 

4? 

6283 

2626 

9734 

8028 

9629 

5634 

7934 

8348 

47.0863 

264.441 

i3 

48 

7419 

3952 

7.01682 

9.20616 

9882 

6174 

8863|26.o3o7 

7395 

286.478 

12 

49 

8559 

5385 

3438 

3oi6 

11.0237 

6719 

9802 

2296 

48.4121 

312.621 

II 

5o 

6.10,703 

6.96823 

7.  96302 

9.26630 

11.0694 

13.7267 

i8.o75o 

26.43i6 

49.1039 

343.774 

10 

61 

6.20861 

8268 

7i76 

8068 

o954 

7821 

i  -708 

6367 

8167 

381.971 

9 

62 

2003 

9718 

9068 

9.30699 

i3i6 

8378 

2677 

845o 

50.5485 

429.718 

8 

53 

3i6o 

7.01174 

8.00948 

3i55 

1681 

8940 

3655 

27.0666 

5i.3o32 

491.106 

7 

54 

43ai 

2637 

2848 

6724 

2048 

9607 

4645 

2716 

62.0807 

672.967 

6 

55 

5486 

4io5 

4756 

83o7 

2417 

14.0079 

5645 

4899 

8821 

687.649 

5 

56 

6655 

6679 

6674 

9.409^4 

2780 

o655 

6656 

7117 

53.7086 

859.436 

4 

57 

7829 

7069 

8600 

35i5 

3i63   1235 

7678 

9372 

54-56i3 

1145.92 

3 

58 

9007 

8546 

8.io536 

6i4i:   354o;   1821 

8711 

28.1664 

55.44i5 

1718.87 

2 

59 

6  3oi89 

7.ioo38 

248  1 

8781'  3919   2411 

9766   3994 

56.35o6 

3437.75 

• 

9° 

8° 

7° 

6° 

5° 

4°     3° 

2° 

1°     0° 

d 

Natural  Co-tangents. 

)T~P  P 

22.  10 

28.46 

37.83 

6.28 

7.88 

iS.oi 

1 

NATURAL    SECANTS. 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

P.  Fait 
to  I'. 

o 

i  .000000 

i  .000004 

i  .000017 

i  ,oooo38 

i  .000068 

i  .000106 

89 

2.5 

I 

000162 

000207 

000271 

ooo343 

000423 

000612 

88 

7.6 

2 

000609 

000716 

ooo83o 

000963 

001084 

001224 

87 

12.7 

3 

001372 

001629 

001696 

001869 

002061 

002242 

86 

I7.8 

A 

002442 

002660 

002867 

003092 

oo3326 

003669 

85 

22.0 

5 

003820 

oo4o8o 

oo4348 

004626 

004911 

006206 

84 

28.? 

6 

006608 

r.  0582O 

oo6i4i 

006470 

006808 

007164 

83 

33.3 

7    007610 

007874 

008247 

008629 

009020 

009419 

82 

38.6 

8 

009828 

OIO245 

o  0671 

011106 

on55o 

OI2OO3 

81 

43.9 

9 

012466 

012936 

o.34i6 

013906 

oi44o3 

014910 

80 

49.3 

10 

1.016427 

I  .016962 

1.016487 

i  .017030 

1.017683 

I.  018145 

79 

54.8 

n 

018717 

019297 

019887 

020487 

021096 

021713 

78 

60.4 

12 

022341 

022977 

023624 

024280 

024946 

026620 

77 

66.0 

i3 

0263o4 

026998 

027702 

028416 

029138 

029871 

76 

71.8 

i4 

o3o6i4 

o3i366 

032128 

©32900 

o33682 

034474 

75 

77-7 

i5 

036276 

o36o88 

036910 

037742 

038584 

039437 

74 

83.7 

16 

040299 

041172 

042066 

042949 

o43853 

044767 

73 

89.8 

J7 

046692 

046627 

047673 

048629 

049496 

060474 

72 

96.1 

18 

061462 

062461 

063471 

064492 

o55524 

066667 

71 

IO2.6 

'? 

067621 

068686 

069762 

060849 

061947 

063067 

7° 

109.2 

20 

1.064178 

i.o653io 

i.o66454 

i  .067609 

1.068776 

I  .069966 

69 

116.1 

21 

071145 

072347 

073661 

074786 

076024 

077273 

68 

123.2 

22 

o78535 

079808 

081094 

082392 

083703 

086026 

67 

i3o.4 

23 

o8636o 

087708 

089068 

090441 

091827 

093226 

66 

137-9 

24 

094636 

096060 

097498 

098948 

ioo4n 

101888 

65 

145.6 

26    103878 

104881 

106398 

107929 

109473 

inoSo 

64 

i53.7 

26  ,   112602 

114187 

116787 

117400 

119028 

120670 

63 

162.0 

27 

122326 

123997 

126682 

127382 

129096 

130826 

62 

170,7 

28 

132670 

134329 

i36io4 

137893 

139698 

141618 

61 

179,7 

29 

143354 

146206 

147073 

148966 

160864 

162769 

60 

189.1 

3o  J  1.164701 

1.  156648 

1.  168612 

i  .  160692 

1.162689 

i.i646o3 

69 

198,9 

3i 

166633 

168681 

170746 

172828 

174927 

177044 

58 

209,  i 

32 

179178 

i8i33i 

i835oi 

186689 

187896 

190120 

67 

219.7 

33    i92363 

194626 

196906 

199206 

201623 

2o386i 

56 

230.9 

34    206218 

208694 

210991 

2i34o6 

216842 

218298 

55 

242.6 

35 

220776 

223271 

226789 

228327 

23o886 

233466 

54 

264.8 

36 

236o68 

238691 

24i336 

244oo3 

246691 

249402 

53 

267.7 

37 

262136 

264892 

267671 

260472 

263298 

266146 

62 

281.3 

38 

269018 

271914 

274834 

277779 

280748 

283741 

5i 

296.6 

39  ;   286760 

289803 

292872 

296967 

299088 

302234 

5o 

3T0.7 

4o  •  i,3o54o7 

i.  308607 

i.3u833 

1.316087 

i.3i8368 

1.321677 

49 

326.7 

4i 

325oi3 

328378 

33i77i 

336192 

338643 

342123 

48 

343.6 

42 

345633 

349172 

362742 

356342 

369972 

363634 

47 

36i.5 

43 

367327 

371062 

374809 

378598 

382420 

386276 

46 

38o.5 

44 

390164 

394o86 

398042 

4O20§2 

406067 

4ion8 

45 

400.7 

45 

4i42i4 

4i8345 

4225l3 

426718 

430960 

435239 

44 

422.3 

46 

439557 

443912 

4483o6 

452740 

467213 

461726 

43 

445.3 

47 

466279 

470874 

476609 

480187 

484907 

489670 

42 

469.8 

48 

494477 

499327 

604221 

609160 

5i4i45 

619176 

4i 

496.2 

49 

624263 

629377 

534549 

539769 

545o38 

55o356 

4o 

524.4 

5o 

i.  666724 

i.  661142 

1.666612 

I.572I34 

i  .677708 

1.583335 

39 

554-7 

5i 

689016 

694761 

600642 

6o6388 

612291 

618261 

38 

587.4 

62 

624269 

63o346 

636483 

642680 

648938 

666268 

37 

622.7 

53 

66i64o 

668086 

674697 

681173 

687816 

694624 

36 

660.9 

54 

701302 

708148 

716064 

722061 

$729110 

73624J 

35 

702.2 

55 

743447 

760727 

768084 

766617 

773029 

780620 

34 

747-2 

56 

788292 

796046 

8o388i 

811801 

819806 

827899 

33 

796.2 

5? 

836o78 

844348 

862707 

861169 

869704 

878344 

32 

849.8 

58 

887080 

896914 

904847 

9i388i 

923017 

932268 

3i 

908.6 

59 

941604 

961068 

960621 

970294 

980081 

989982 

3o 

973.o 

60' 

50' 

40' 

30' 

20' 

10' 

Deg. 

Natural  Co-secants. 


J\  \TURAL    SECANTS. 


135 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

P.  Part 
tol'. 

60 

2.OOOOOO 

2.oioi36 

2  .020393 

2.030772 

2.041276 

2.061906 

29 

io44 

61    062666 

o73556 

084579 

096739 

107036 

118474 

28 

1123 

'  62 

i3co54 

141781 

i53655 

166681 

177869 

190196 

27 

I2IO 

63 

202685 

2i5346 

22$l68 

241168 

264320 

267667 

26 

1808 

64    281172    294869 

308760 

322820 

337083 

35i542 

25 

1417 

65 

366202 

38io65 

396137 

4n42i 

426922 

442645 

24 

1689 

66 

458593 

474773 

491187 

607843 

324744 

541896 

28 

1678 

67 

5593o5 

676976 

694914 

6i3i26 

53i6i8 

660396 

22 

i835 

68 

669467 

688387 

708614 

728604 

748814 

769453 

21 

2016 

69 

790428 

811747 

8334i9 

85545i 

877853 

900635 

2O 

2222 

7c 

2.923804 

2.947372 

2.971349 

2.996744 

3.020669 

3.045835 

19 

246l 

71 

3.o7i553 

3.097736 

3.124396 

3.i5i545 

179198 

207867 

18 

2740 

72 

236o68 

2653i5 

296123 

326610 

356490 

388o82 

17 

3o68 

73 

42o3o4 

453i73 

486711 

620987 

55587i 

59i536 

16 

3458 

74 

627966 

666162 

7o3i5i 

741978 

781660 

822226 

i5 

8926 

75 

8637o3 

906126 

9^9622 

993929 

4.039380 

4.086918 

14 

4492 

76 

4.133565 

4.182378 

4.282894 

4.283658 

3362i5 

890116 

18 

6190 

77 

4454H 

602167 

56o4o8 

620226 

681676 

744821 

12 

6062 

78 

809734 

876491 

946169 

6.016862 

6.088628 

5.i63592 

II 

7171 

79 

5.  24o843 

5.32o486 

5.402633 

487404 

674926 

665333 

IO 

8612 

80 

6.768770 

5.855392 

5.955362 

6.068868 

6.166067 

6.277198 

9 

81 

6.392453 

6.  612081 

6.636329 

6.766469 

6.899794 

7.089622 

8 

82 

7.  186297 

7.337191 

7.496711 

7.661298 

7.834433 

8.015645 

7 

83 

8.206609 

8.404669 

8.613790 

8.83367i 

9.066161 

9.809170 

6 

84 

9.666772 

9.839123 

10.  12762 

10.43343 

10.76849 

i  i.  io455 

5 

85 

11.47371 

11.86837 

12.29126 

12.74549 

13.23472 

18.76811 

4 

86 

14.33669 

14.96788 

i5.63679 

i6.38o4i 

17.19843 

18.  10262 

3 

i  87 

19.  10782 

20.23028 

21.49368 

22.92669 

24.66212 

26.46061 

2 

88 

28.66371 

31.26768 

34.38232 

38.2oi55 

42.97671 

49.  n4o6 

I 

89 

67.29869 

68.75736 

85.9456i 

114.6930 

171.  8883 

343.7762 

O 

60' 

50' 

40' 

30' 

20' 

10' 

Deg. 

Natural  Co-secants. 

LENGTHS  OF  CIRCULAR  ARCS. 

Degrees. 

Minutes. 

Seconds. 

o 

I 

.0174533 

0 

26 

.4537866 

0 

61 

.8901179 

I 

.0002909 

I 

.OOOOO48 

2 

.0349066 

27 

.4712389 

62 

.9076712 

2 

.0006818 

2 

.0000097 

3 

.0623699 

28 

.4886922 

53 

.9260246 

3 

.0008727 

3 

.0000145 

4 

.0698132 

29 

.5o6i455 

54 

.9424778 

4 

.ooii636 

4 

.0000194 

5 

.0872666 

3o 

.6236988 

55 

.9699311 

5 

.0014544 

5 

.0000242 

6 

.  1047198 

3i 

.5410621 

56 

•9773844 

6 

.0017453 

6 

.0000291 

7 

.  1221730 

32 

.5585o54 

67 

•9948377 

7 

.0020362 

7 

.OOOoSSg 

8 

.1396263 

33 

.6769687 

58 

.0122910 

8 

.0023271 

8 

.oooo388 

9 

.  1670796 

34 

.6934119 

59 

.0297443 

9 

.0026180 

9 

.oooo436 

10 

.1745329 

35 

.6108662 

60 

.0471976 

10 

.0029089 

10 

.0000486 

ii 

.1919862 

36 

.6283i85 

65 

.  i  34464o 

ii 

.0031998 

n 

.oooo533 

12 

.2094396 

37 

.6467718 

70 

.2217306 

12 

.0034907 

12 

.0000682 

i3 

.2268928 

38 

.6632261 

75 

.3089969 

i3 

.0037816 

i3 

.oooo63o 

i4 

.244346i 

39 

.6806784 

80 

.3962634 

i4 

.0040724 

i4 

.0000679 

16 
16 

.2617994 
.2792627 

4o 
4i 

.6981317 
.7166860 

85 
9° 

.4836299 
.6707963 

i5 
16 

.oo43633 
.0046642 

16 
16 

.0000727 
.0000776 

17 

.2967060 

42 

.733o383 

IOO 

.7453293 

'7 

.0049461 

17 

.0000824 

1.8 

.3i4i593 

43 

.7604916 

no 

.9198622 

18 

.0062360 

18 

.0000878 

J9 

.33i6i26 

44 

.7679449 

120 

2.0943961 

19 

.0066269 

19 

.0000921 

20 

.3490669 

45 

.7863982 

i3o 

2.2689280 

20 

.0068178 

20 

.0000970 

21 

.3665i9i 

46 

.8028616 

i4o 

2.4434610 

25 

.0072722 

25 

.OOOI2I2 

22 

.3839724 

47 

.8203047 

160 

2.6179939 

3o 

.0087266 

3o 

.0001454 

23 

.4014267 

48 

.8377680 

160 

2.7926268 

4o 

.0116355 

4o 

.0001989 

24 

.4188790 

49 

.8552113 

170 

2.9670697 

5o 

.0145444 

5o 

0002424 

25 

.4363323 

5o 

.8726646 

180 

3.  1416927 

60 

.0174533 

60 

0002909  ! 

130 


TRAVERSE    TABLE. 


Course. 

Dist.  1. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  6. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1  Lat. 

Dep. 

Lat. 

Dep. 

1 

o   / 

I  o  '  ; 

o  i5 

I.OOOO 

<xoo4^ 

?  0000 

0.0087 

3.  OOOO 

O.OlSl 

4-oooc 

0.0175 

5.  oooo 

0.0218 

,'89  45 

3o 

oooo 

0087 

1.9999 

OI75 

2.9999 

0262 

3.999S 

o349 

4.9998 

o436 

'  0 
3o 

45 

0.9999 

oi3i 

9998 

0262 

9997 

o393 

9997 

O522 

9996 

o654 

i5 

I    0 

9998 

0175 

9997 

o349 

9995 

o524 

9994 

o698 

9992 

o873 

89   o 

i5 

9998 

0218 

9995 

o436 

9993 

o654 

999° 

0873 

9988 

io9i 

45 

3o 

9997 

0262 

9993 

o524 

999° 

o785 

9986 

1047 

9983 

i3o9 

3o 

45 

r 

9995 

o3o5 

9991 

0611 

9986 

o9i6 

998i 

1222 

9977 

l527 

i5 

2    0 

9994 

o349 

9988 

0698 

9982 

io47 

9976 

i396 

997° 

i745 

88   o 

i5 

9992 

o39: 

9985 

o785 

9977 

1178 

9969 

1570 

996i 

i963 

45 

3o 

999° 

o436 

9981 

o872 

997i 

i3o9 

9962 

1  745 

9952 

2181 

3o 

45 

0.9988 

o.o48o 

1.9977 

0.0060 

2.9966 

o.i439 

3.9954 

o.i9i9 

4-9942 

0.2399 

i5 

3   o 

9986 

o523 

9973 

io47 

9959 

i57o 

9945 

209: 

993i 

26I? 

87   o 

i5 

9984 

0567 

9968 

1  1  34 

9952 

1701 

9936 

2268 

9920 

2835 

45 

3o 

9981 

0610 

9963 

1221 

9944 

i83i 

9925 

2442 

99°7 

3o52 

3o 

45 

9979 

o654 

9957 

i3o8 

9936 

I962 

9914 

2616 

9893 

3270 

i5 

4  o 

9976 

0698 

9951 

i395 

9927 

2O93 

99o3 

2790 

9878 

3488 

86   o 

i5 

9973 

0741 

9945 

1482 

99i8 

2223 

9890 

2964 

9863 

37o5 

45 

3o 

9969 

o785 

9938 

!569 

99o8 

2354 

9877 

3i38 

9846 

3923 

3o 

45 

9966 

0828 

993i 

i656 

9897 

2484 

9863 

33  1  2 

9828 

4i4o 

i5 

£   o 

9962 

0872 

9924 

I743 

9886 

26i5 

9848 

3486 

98io 

4358 

85   o 

i5 

o.9958 

o.ogiS 

1.9916 

o.i83o 

2.9874 

o.2745 

3.9832 

o.366o 

4.979o 

o.4575 

45 

3o 

9954 

0958 

9908 

1917 

9862 

2875 

9816 

3834 

977° 

4792 

3o 

45 

995° 

1  002 

9899 

2O04 

9849 

3oo6 

9799 

4oo8 

9748 

5009 

i5 

6  o 

9945 

io45 

9890 

2091 

9836 

3i36 

978i 

4i8i 

9726 

5226 

84   o 

i5 

9941 

1089 

9881 

2I77 

9822 

3266 

9762 

4355 

97°3 

5443 

45 

3o 

9936 

11*32 

9871 

2264 

9807 

3396 

9743 

4528 

9679 

566o 

3o 

45 

993i 

1175 

9861 

235i' 

9792 

3526 

9723 

47oi 

9653 

5877 

i5 

7   o 

9925 

1219 

985i 

2437 

9776 

3656 

0.702 

4875 

9627 

6o93 

83   o 

i5 

9920 

1262 

9840 

2524 

9760 

3786 

968o 

5o48 

96oo 

63io 

45 

3o 

99i4 

i3o5 

9829 

2611 

9743 

39i6 

9658 

5221 

9572 

6526 

3o 

45 

0.9909 

o.i349 

1.9817 

o.2697 

2.9726 

o.4o46 

3.9635 

o.5394 

4-9543 

o.6743 

i5 

8   o 

99o3 

i392 

9805 

2783 

9708 

4i75 

96n 

5567 

95i3 

6959 

82   o 

i5 

9897 

i435 

9793 

2870 

9690 

43o5 

9586 

574o 

9483 

7i75 

45 

3o 

9890 

1478 

9780 

2956 

9670 

4434 

956i 

59I2 

c45i 

739o 

3o 

45 

9884 

l52I 

9767 

3o42 

965i 

4564 

9534 

6o85 

94i8 

76o6 

i5 

9  ° 

9877 

1  564 

9754 

3129 

963i 

4693 

95o8 

6257 

9^84 

7822 

81   o 

i5 

9870 

1607 

974o 

32  1  5 

9610 

4822 

948o 

643o 

q35o 

8o37 

45 

3o 

9863 

i65o 

9726 

33oi 

9589 

495i 

945i 

6602 

93i4 

8252 

3o 

45 

9856 

1693 

9711 

3387 

9567 

5o8o 

9422 

6774 

9278 

8467 

i5 

10   o 

9848 

1736 

9696 

3473 

9544 

52O9 

9392 

6946 

924o 

8682 

80   o 

i5 

0.9840 

0.1779 

1.9681 

o.3559 

2.952I 

0.5338 

3.9362 

o.7n8 

i.92O2 

0.8807 

45 

3o 

9833 

1822 

9665 

3645 

9498 

5467 

933o 

7289 

9  1  63 

9II2 

3o 

45 

9825 

i865 

9649 

373o 

9474 

5596 

9298 

746i 

9I23 

9326 

i5 

Jl    0, 

9816 

1908 

9633 

38i6 

9449 

5724 

9265 

7632 

9o8i 

954o 

79   ° 

i5 

9808 

i95i 

9616 

3902 

94a4 

5853 

923l 

7804 

9o39 

9755 

45 

3o 

9799 

1994 

9598 

3987 

9398 

598i 

0107 

7975 

8996 

9968 

3o 

45 

979° 

2o36 

958i 

4o73 

937i 

6io9 

9l62 

8i46 

8952 

1.0182 

1  5 

12   o 

9781 

2079 

9563 

4i58 

9344 

6237 

9I26 

83i6 

89o7 

o396 

78   o 

i5 

9772 

2122 

9545 

4244 

93i7 

6365 

9o89 

8487 

8862 

o6o9 

45 

3o 

9763 

2164 

9.526 

4329 

9289 

6493 

9OD2 

8658 

88i5 

0822 

3o 

45 

o.9753 

0.2207 

1.9507 

o.44i4 

.926o 

0.6621 

3.9oi  \ 

0.8828 

4.8767 

io35 

i5 

i3   o 

9744 

225o 

9487 

4499 

923l 

6749 

897S 

8998 

8719 

1248 

77   ° 

i5 

9734 

2292 

9468 

4584 

920I 

6876 

8935 

9i68 

8669 

r46o 

45 

3o 

9724 

2334 

944? 

4669 

9I7I 

7oo3 

8895 

9338 

8618 

1672 

3o 

45 

97i3 

2377 

9427 

4754 

9i4o 

7i3i 

8854 

95°7 

8567 

1884 

-5 

i4  o 

97o3 

2419 

94o6 

4838 

9I09 

7258 

8812 

9677 

85i5 

2O96 

76   o 

i5 

9692 

2462 

9385 

4923 

9°77 

7385 

8769 

9846 

8462 

23o8 

A5 

3o 

9681 

25o4 

9363 

5oo8 

9044 

75n 

8726 

i.ooiS 

84o7 

a5i9 

3o 

45 

9670 

2546 

934i 

5092 

9011 

7638 

8682 

0184 

8352 

2730 

i5 

i5   o 

9659 

2588 

93i9 

5i76 

8978 

7765 

8687 

o353 

8296 

294l 

75   o 

• 

Dep. 

Lat. 

Dep.    Lat. 

Dep.  |  Lat. 

Dep. 

Lat.  ! 

.Dep. 

Lat. 

Diet.  1. 

Dist.  2. 
| 

Dist.  3. 

Dist.  4.  | 

Dist.  5. 

C  our  ae.  | 

TRAVERSE    TABLE. 


137 


Course. 

Dist,  0. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

Lat. 

Dcp. 

Lnt. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

"  o   / 

0      / 

o  i5 

6.9999 

0.0262 

6.9999 

o.o3o5 

7-9999 

o.o349 

8.9999 

o.o393 

9.9999 

o.o436 

89  45 

3o 

9998 

o524 

9997 

0611 

9997 

0698 

9997 

o785 

9996 

o873 

3o 

45 

9995 

0785 

9994 

o9i6 

9993 

io47 

9992 

1178 

999I 

i3oq 

i5 

:   o 

9991 

1047 

9989 

1222 

9988 

1396 

9986 

1671 

9985 

17** 

8a   o 

i5 

9986 

1  309 

9983 

1527 

998i 

i745 

9979 

1963 

9976 

2181 

45 

3o 

9979 

1571 

9976 

i832 

9973 

2094 

9969 

2356 

9966 

2618 

3o 

45 

9972 

i832 

9967 

2i38 

9963 

2443 

9958 

2748 

9953 

3o54 

i5 

a   o 

9963 

2094 

9957 

2443 

995i 

2792 

9945 

3i4i 

9939 

349o 

88   o 

i5 

9954 

2356 

9946 

2748 

9938 

3i4i 

993i 

3533 

9923 

3926 

45 

3o 

9943 

2617 

9933 

3o53 

9924 

3490 

99i4 

3926 

99o5 

4362 

3o 

45 

5.993i 

0.2879 

6.9919 

0.3358 

7.99o8 

0.3838 

8.9896 

o.43i-8 

9.9885 

o.4798 

i5 

3   o 

9918 

3i4o 

9904 

3664 

989o 

4i87 

9877 

47io 

9863 

5234 

87   o 

i5 

9904 

3402 

9887 

3968 

987i 

4535 

9855 

6102 

9839 

5669 

45 

3o 

9888 

3663 

9869 

4273 

985i 

4884 

9832 

5494 

98i3 

6io5 

3o 

45 

9872 

3924 

985o 

4578 

9829 

5232 

98o7 

5886 

9786 

654o 

i5 

4  o 

9854 

4i85 

9829 

4883 

98o5 

558i 

978i 

6278 

9756 

6976 

86   o 

i5 

9835 

444? 

98o8 

5i88 

978o 

5929 

9753 

6670 

9725 

74n 

45 

3o 

9815 

4708 

9784 

5492 

9753 

6277 

9723 

7061 

9692 

7846 

3o 

45 

9794 

4968 

9760 

5797 

9725 

6625 

969i 

7453 

9657 

8281 

i5 

5   o 

9772 

5229 

9734 

6101 

9696 

6972 

9658 

7844 

9619 

8716 

85   o 

i5 

5.9748 

0.5490 

6.9706 

o.64o5 

7.9664 

0.7320 

8.9622 

o.8235 

9.958o 

o.9i5o 

45 

3o 

9724 

575i 

9678 

67o9 

9632 

7668 

9586 

8626 

954o 

9585 

3o 

45 

9698 

6011 

9648 

7oi3 

9597 

8oi5 

9547 

9017 

9497 

1.0019 

i5 

(5   o 

9671 

6272 

9617 

73i7 

9562 

8362 

95°7 

94o8 

9452 

o453 

84   o 

x5 

9643 

6532 

9584 

762I 

9525 

8709 

9465 

9798 

94o6 

0887 

45 

3c 

9614 

6792 

955o 

7924 

9486 

9o56 

942I 

1.0188 

9357 

1320 

3o 

45 

9584 

7062 

95i5 

8228 

9445 

94o3 

9376 

0678 

93o7 

1754 

i5 

7   o 

9553 

73l2 

9478 

853i 

94o4 

975o 

9329 

o968 

9255 

2187 

83   o 

i5 

9520 

7572 

944o 

8834 

936o 

1.0096 

928o 

i358 

92OO 

2620 

45 

3o 

9487 

7832 

94oi 

9i37 

93i6 

0442 

923o 

1747 

9i44 

3o53 

3o 

45 

5.9452 

0.8091 

6.936i 

o.944o 

7.9269 

1.0788 

8.9i78 

i.ai37 

9.9o87 

i  3485 

i5 

8   o 

94i6 

835o 

93i9 

9742 

9221 

n34 

9I24 

2626 

9027 

39i7 

82   o 

:5 

9379 

8610 

9276 

i.oo44 

9I72 

1479 

9o69 

29l4 

8965 

4349 

45 

3o 

934i 

8869 

923l 

o347 

9I2I 

1825 

9on 

33o3 

8902 

478i 

3o 

45 

93o2 

9:27 

9i85 

o649 

9069 

2170 

8953 

369i 

8836 

5212 

i5 

9  ° 

926l 

9386 

9i38 

o95o 

9oi5 

25i5 

8892 

4o79 

8769 

5643 

81   o 

i5 

922O 

9645 

9o9o 

1252 

8960 

2859 

883o 

4467 

8700 

6o74 

45 

3o 

9i77 

9903 

9o4o 

i553 

8903 

3204 

8766 

4854 

8629 

65o5 

3o 

45 

9i33 

1.0161 

8989 

i854 

8844 

3548 

87oo 

6241 

8556 

6935 

i5 

1(5   0 

9088 

0419 

8937 

2i55 

8785 

3892 

8633 

6628 

848  1 

7365 

80   o 

i5 

5.9042 

1.0677 

6.8883 

1.2456 

7.8723 

1.4235 

8.8564 

i.6oi5 

9.84o4 

i.7794 

45 

3o 

8995 

0934 

8828 

2756 

8660 

4579 

8493 

64oi 

8325 

8224 

3b 

45 

8947 

1191 

8772 

3o57 

8596 

4922 

8421 

6787 

8245 

8652 

i5 

11    O 

8898 

i44g 

87i4 

3357 

853o 

5265 

8346 

7i73 

8i63 

9o8i 

79   ° 

i5 

8847 

1705 

8655 

3656 

8463 

56o7 

827I 

7558 

8o79 

95o9 

45 

3o 

8795 

1962 

8595 

3956 

8394 

5949 

8i93 

7943 

7992 

9937 

3o 

45 

8743 

2219 

8533 

4255 

8324 

6291 

8n4 

8328 

79°5 

2.o364 

i5 

12    0 

8689 

2475 

8470 

4554 

8252 

6633 

8o33 

8712 

78i5 

O79i 

78   o 

i5 

8634 

273l 

84o6 

4852 

8178 

6974 

795i 

9o96 

7723 

1218 

45 

3o 

8578 

2986 

834i 

5i5i 

8io4 

73i5 

7867 

948o 

763o 

1  644 

3o 

45 

5.8521 

1.3242 

6.8274 

i.5449 

7.8027 

1.7666 

8.7781 

i.9863 

9.7534 

2.2070 

i5 

i3  o 

8462 

3497 

8206 

5747 

795o 

•7996 

7693 

2.0246 

7437 

2495 

77   ° 

i5 

84o3 

3752 

8i37 

6o44 

787o 

8336 

7602 

0628 

7338 

292O 

45 

3o 

8342 

4oo7 

8066 

634i 

779° 

8676 

75i3 

IOIO 

7237 

3345 

3o 

45 

8281 

4261 

7994 

6638 

77°7 

9oi5 

7421 

i392 

7i34 

3769 

i5 

i4  o 

8218 

45i5 

792I 

6935 

762^ 

9354 

7327 

1773 

7o3o 

4l92 

76   o 

i5 

8i54 

4769 

?84C 

723l 

7538 

9692 

723l 

2162 

6923 

46i5 

45 

3o 

8089 

5o2: 

777° 

7527 

7452 

2.oo3o 

7i33 

253; 

68i5 

5o38 

3o 

45 

802^ 

62-76 

7693 

7822 

7364 

o368 

7o34 

29I2 

67o5 

546o 

i5 

i5   o 

7956 

5529 

76i5 

8117 

7274 

0-706 

6933 

329/ 

6593 

6882 

75   o 

Dep. 

Lat 

De>». 

Lat. 

Dep. 

Lat. 

Dep.    Lat. 

Dep. 

Lat 

Dist.  6. 

JDist.  7. 

Dist.  8. 

D*5t.  9.    Dist.  10. 

Course. 

!  3  fl 


TRAVERSE    T  A  B  L  K. 


Course. 

Dist.  t. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

J 

o   / 

i5  i5 

0-9648 

0.2630 

1.9296 

0.526l 

2.8944 

o.789i 

3.859i 

I.O52I 

48239 

I.3l52 

o   7^ 
74  45 

3o 

9636 

2672 

9273 

5345 

89o9 

8oi7 

8545 

0690 

$182 

3362 

3o 

45 

96*25 

2714 

9249 

5429 

8874 

8i43 

8498 

o858 

8i23 

3572 

i5 

16   o 

9613 

2756 

9226 

55i3 

8838 

8269 

845o 

1025 

8o63 

3782 

74   o 

i5 

9600 

2798 

9201 

5597 

8801 

8395 

8402 

1193 

8002 

3991 

45 

3o 

9588 

2840 

9176 

568o 

8765 

8520 

8353 

i36i 

794i 

4201 

3o 

45 

9576 

2882 

9i5i 

5764 

8727 

8646 

83o3 

i528 

7879 

44io 

i5 

17   o 

9563 

292; 

9126 

5847 

8689 

8771 

8252 

1695 

78i5 

46i9 

73   o 

i5 

955o 

2965 

9100 

593i 

865i 

8896 

8201 

1862 

775i 

482? 

45 

3o 

9537 

3007 

9074 

6012 

8612 

9021 

8i49 

2028 

7686 

5o35 

3o 

45 

0.9522 

0.3049 

1.9048 

0.6097 

2.8572 

0.9146 

3.8o96 

1.2195 

4.7620 

1.5243 

i5 

i9   o 

95u 

Sogo 

9021 

6180 

8532 

9271 

8042 

236i 

7553 

545i 

72   o 

i5 

9497 

3i32 

8994 

6263 

8491 

9395 

7988 

2527 

7485 

5658 

45 

3o 

9483 

3i73 

8966 

6346 

845o 

95i9 

7933 

2692 

7416 

5865 

3o 

45 

9469 

32  1  4 

8939 

6429 

84o8 

9643 

7877 

2858 

7347 

6o72 

i5 

19   o 

9455 

3256 

8910 

65n 

8366 

•  9767 

7821 

3o23 

7276 

6278 

71   o 

i5 

944  1 

3297 

8882 

6594 

8323 

9891 

7764 

3i88 

7202 

6485 

45 

3o 

9426 

3338 

8853 

6676 

8279 

I.OOI^ 

7706 

3352 

7l32 

6690 

3o 

45 

9412 

3379 

8824 

6758 

8235 

oi38 

7647 

35i7 

7o59 

6896 

i5 

20   o 

9397 

3420 

8794 

684o 

8191 

0261 

7588 

368i 

6985 

7101 

70   o 

i5 

o.9382 

i.  346i 

1.8764 

0.6922 

2.8146 

i.o384 

3.7528 

1.3845 

4.6910 

1.7306 

45 

3o 

9367 

35o2 

8733 

7004 

8100 

o5o6 

7467 

4oo8 

6834 

75io 

3o 

45 

935i 

3543 

8703 

7086 

8o54 

0629 

74o5 

4l72 

6757 

77i5 

i5 

21    O 

9336 

3584 

8672 

7167 

8007 

0751 

7343 

4335 

6679 

•7918 

59   o 

i5 

9320 

3624 

864o 

7249 

7960 

0873 

7280 

4498 

6600 

8122 

45 

3o 

93o4 

3665 

8608 

733o 

79i3 

0995 

7217 

466o 

652i 

8325 

3o 

45 

9288 

37o6 

8576 

74" 

7864 

1117 

7i52 

4822 

644o 

8528 

i5 

22   0 

9272 

3746 

8544 

7492 

7816 

1238 

7087 

4984 

6359 

873o 

68   o 

i5 

9255 

3786 

85u 

7573 

7766 

i359 

7022 

5:46 

6277 

8932 

45 

3o 

9239 

3827 

8478 

7654 

7716 

i48i 

6955 

53o7 

6i94 

9i34 

3o 

45 

0.9222 

o.3867 

1.8444 

0.7734 

2.7666 

1.1601 

3.6888 

1.5468 

i.6no 

1.9336 

i5 

*3   o 

9205 

39o7 

84io 

78:5 

76i5 

1722 

6820 

5629 

6025 

9537 

67   o 

i5 

9188 

3947 

8376 

7895 

7564 

1842 

6752 

579o 

594o 

9737 

45 

3o 

9171 

3987 

834i 

7975 

7512 

1962 

6682 

595o 

5853 

9937 

3o 

45 

9i53 

4027 

83o6 

8o55 

7459 

2082 

6612 

6110 

5766 

2.0137 

i5 

a4   o 

9i35 

4067 

8271 

8i35 

7406 

2202 

6542 

6269 

5677 

o337 

66   o 

:5 

9u8 

4107 

8235 

8214 

7353 

2322 

6470 

6429 

5588 

o536 

45 

3o 

9100 

4i47 

8199 

8294 

7299 

2441 

6398 

6588 

5498 

o735 

3o 

45 

9081 

4i87 

8i63 

8373 

7244 

256o 

6326 

6746 

54o7 

0933 

i5 

25   0 

9063 

4226 

8126 

8452 

7i89 

2679 

6252 

6905 

53i5 

n3i 

65   o 

i5 

0.9045 

0.4266 

1.8089 

o.853i 

2.7i34 

1.2797 

3.6178 

1.7063 

..5z23 

2.1328 

45 

3o 

'9026 

43o5 

8o52 

8610 

7o78 

2916 

6io3 

7220 

5l29 

i526 

3o 

45 

9007 

4344 

8014 

8689 

702I 

3o33 

6028 

7378 

5o35 

1722 

i5 

26   o 

8988 

4384 

7976 

8767 

6964 

3i5i 

5952 

7535J 

494o 

1919 

64   o 

i5 

8969 

4423 

7937 

8846 

6906 

3269 

5875 

7692 

4844 

2Il4 

45 

3o 

8949 

4462 

7899 

8924 

6848 

3386 

5797 

7848 

4747 

23lO 

3o 

45 

8930 

45oi 

7860 

9002 

6789 

35o3 

57i9 

8oo4 

4649 

25o5 

i5 

27  o 

8910 

454o 

7820 

9080 

673o 

3620 

564o 

8160 

455o 

2700 

63   o 

i5 

8890 

4579 

7780 

9l57 

667i 

3736 

556i 

83i5 

445  1 

2894 

45 

3o 

8870 

46i7 

7740 

9235 

6610 

3852 

548o 

847o 

435  1 

3o87 

3o 

£5 

£85o 

o.4656 

7700 

0.9312 

.655o 

i.3968 

3.54oo 

1.8625 

4.4249 

.8281 

i5 

28   o 

8829 

4695 

7659 

9389 

6488 

4o84 

53i8 

8779 

4i47 

3474 

62   o  ' 

i5 

8809 

4733 

7618 

9466 

6427 

4200 

5236 

8933 

4o45 

3666 

45, 

3o 

8788 

4772 

7576 

9543 

6365 

43i5 

5i53 

9086 

3941 

3858 

3o 

45 

8767 

48io 

7535 

9620 

63o2 

443o 

5069 

9240 

3836 

4o4g 

16 

9O    0 

8746 

4848 

7492 

9696 

6239 

4544 

4985 

9392 

373i 

4240 

61   o 

i5 

8726 

4886 

745o 

9772 

6i75 

4659 

4900 

9545 

3625 

443  1 

45 

3o 

8704 

4924 

7407 

9848 

6111 

4773 

48i4 

9697j 

35i8 

4621 

3o 

45 

8682 

496a 

7364 

9924 

6o46 

4886 

4728 

9849i 

34io 

48u 

16 

3o   o 

8660 

5ooo 

7321 

.0000 

598i 

5ooo 

464i 

2.0000 

33oi 

5ooo 

60   o 

Dcp. 

Lnt 

Dep. 

Lat. 

Dep.    Lat. 

Dep. 

Lnt. 

Pep. 

Lat 

1 

Dist.  1. 

Dist.  2. 

Dist.  3. 

Dist.  4.     Dist.  5. 

Course.  | 



U  

"  L  ' 

TRAVERSE    TABLE. 


Course. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

| 

Lat 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

o   / 

o    / 

i5  i5 

5.7887 

1.5782 

6.7535 

I.84I2 

7.7i83 

2.IO42 

8.683i 

2.3673 

9.6479 

2.63o3 

74  45 

3o 

7818 

6o34 

7454 

8707 

7°9° 

i379 

6727 

4o5i 

6363 

6724 

3o 

45 

7747 

6286 

7372 

9001 

6996 

1715 

6621 

443o 

6246 

7144 

i5 

16   o 

7676 

6538 

7288 

9295 

6901 

2o5i 

65i4 

4807 

6126 

7564 

74   o 

i5 

76o3 

6790 

7203 

9588 

68o4 

2386 

64o4 

5  1  85 

6oo5 

7983 

45 

3o 

7529 

7041 

7117 

988i 

6706 

2721 

6294 

556i 

5882 

8402 

3o 

45 

7454 

7292 

7o3o 

2.0174 

6606 

3o56 

6181 

5938 

5757 

8820 

i5 

17   o 

7378 

7542 

6g4i 

o466 

65o4 

339o 

6067 

63i3 

563o 

9237 

73   o 

i5 

73oi 

7792 

685: 

0758 

6402 

3723 

5952 

6689 

55o2 

9654 

45 

3o 

7223 

8042 

6760 

io49 

6297 

4o56 

5835 

7o64 

5372 

3.0071 

3o 

45 

5.7i44 

1.8292 

6.6668 

2.l34l 

7.6192 

2.4389 

8.57i6 

2.7438 

9.5240 

3.o486 

i5 

18   o 

7063 

854i 

6574 

i63i 

6o85 

4721 

5595 

7812 

5io6 

O9O2 

72   o 

i5 

6982 

8790 

6479 

I92I 

5976 

5o53 

5473 

8i85 

4970 

i3i6 

45 

3o 

6899 

9o38 

6383 

2211 

5866 

5384 

5349 

8557 

4832 

1730 

3o 

45 

6816 

9286 

6285 

25oi 

5754 

57i5 

5224 

8930 

4693 

2i44 

i5 

10    O 

6731 

9534 

6186 

2790 

564i 

6o45 

5o97 

9301 

4552 

2557 

71   o 

i5 

6645 

9781 

6086 

3o78 

5527 

6375 

4968 

9672 

4409 

2969 

45 

3o 

6558 

2.0028 

5985 

3366 

54u 

6705 

4838 

3.oo43 

4264 

338i 

3o 

45 

6471 

0275 

5882 

3654 

5294 

7o33 

4706 

o4i3 

4n8 

3792 

i5 

20   o 

6382 

O52I 

5778 

394i 

,  5l75 

7362 

4572 

0782 

3969 

4202 

70   o 

i5 

5.6291 

2.0767 

6.5673 

2.4228 

7.5o55 

2.7689 

8.4437 

3.u5i 

9.38i9 

3.46i2 

45 

3o 

6200 

1012 

5567 

45i5 

4934 

8017 

43oo 

i5i9 

3667 

5O2I 

3o 

45 

6108 

1257 

5459 

48oo 

48n 

8343 

4162 

1886 

35i4 

5429 

i5 

fcl    0 

6oi5 

i5o2 

535i 

5o86 

4686 

8669 

4O22 

2253 

3358 

5837 

60   o 

i5 

5920 

1746 

524i 

5371 

456i 

8995 

388i 

26i9 

3201 

6244 

45 

3o 

5825 

1990 

5l29 

5655 

4433 

932O 

3738 

2985 

-  3o42 

665o 

3o 

45 

5729 

2233 

5017 

5939 

43o5 

9645 

3593 

335o 

2881 

7o56 

i5 

22    O 

563i 

2476 

49o3 

6222 

4i75 

9969 

3447 

37i5 

2718 

746  1 

68   o 

i5 

5532 

2719 

4788 

65o5 

4o43 

3.O292 

3299 

4o78 

2554 

7865 

45 

3o 

5433 

2961 

4672 

6788 

39io 

o6i5 

3i49 

4442 

2388 

8268 

3D 

45 

5.5332 

2.3203 

6.4554 

2.7070 

7.3776 

3.o937 

8.2998 

3.48o4 

9.2220 

3.8671 

i5 

23   0 

523o 

3444 

4435 

735i 

364o 

1258 

2845 

5i66 

2o5o 

9073 

67   o 

i5 

5127 

3685 

43i5 

7632 

35o3 

i58o 

269I 

5527 

1879 

9-474 

45 

3o 

5o24 

3920 

4i94 

79I2 

3365 

1900 

2535 

5887 

1706 

9875 

3o 

45 

4919 

4i65 

4071; 

8l92 

3225 

222O 

2378 

6247 

i53i 

4.0276 

i5 

24   o 

48i3 

44o4 

3948 

8472 

3o84 

2539 

22I9 

6606 

i355 

0674 

66   o 

i5 

4706 

4643 

3823 

875o 

294i 

2868 

2o59 

6965 

1176 

1072 

45 

3o 

4598 

4882 

3697 

9O29 

2797 

3175 

l897 

7322 

0996 

i469 

3o 

45 

4489 

5l20 

3570 

93o6 

265i 

3493 

I733 

7679 

0814 

1866 

i5 

25   0 

4378 

5357 

3442 

9583 

25o5 

3809 

i568 

8o36 

o63i 

2262 

65   o 

i5 

5.4267 

2.5594 

6.33i2 

2.986o 

7.2356 

3.4i25 

8.i4oi 

3.839i 

9.0446 

4.2657 

45 

3o 

4i55 

583i 

3i8i 

3.oi36 

2207 

444  1 

1233 

8746 

0259 

3o5: 

3o 

45 

4042 

6067 

3o49 

o4n 

2o56 

4756 

io63 

9ioo 

0070 

3445 

i5 

26   o 

3928 

63o2 

2916 

0686 

I9o4 

5070 

0891 

9453 

8.9879 

3837 

64   o 

i5 

38i2 

6537 

2781 

o96o 

1750 

5383 

0719 

98o6 

9687 

4229 

45 

3o 

3696 

6772 

2645 

1234 

i595 

5696 

o544 

4.oi58 

9493 

4620 

3o 

45 

3579 

7006 

2509 

1507 

i438 

6008 

o368 

o5o9 

9298 

5oio 

i5 

27   o 

346o 

7239 

2370 

1779 

1281 

63i9 

0191 

o859 

9101 

5399 

63   o 

i5 

334i 

7472 

223l 

2o5i 

II2I 

663o 

OOI2 

I2O9 

8902 

5787 

45 

3o 

3221 

77°5 

2091 

2322 

o96i 

6g4o 

7.983i 

i557 

8701 

6i75 

3o 

45 

5.3099 

2.7937 

6.1949 

3.2593 

7-°799 

3.7249 

7.9649 

4-i9o5 

8.8499 

4.656i 

i5 

28   o 

2977 

8168 

1806 

2863 

o636 

7558 

9465 

2252 

8295 

6947 

62   o 

i5 

2853 

8399 

1662 

3i32 

0471 

7866 

9280 

2599 

8089 

7332 

45 

3o 

2729 

863o 

i5i7 

34oi 

o3o5 

8i73 

9094 

2944 

7882 

7716 

3o 

45 

2604 

8859 

i37i 

3669 

oi38 

8479 

8906 

3289 

7673 

8o99 

i5 

20   0 

2477 

9089 

1223 

3937 

6.997o 

8785 

8716 

3633 

7462 

848i 

61   o 

i5 

235o 

93i7 

IO75 

4203 

98oo 

9090 

8525 

3976 

725o 

8862 

45 

3o 

2221 

9545 

0925 

4470 

9628 

9394 

8332 

43i8 

7o36 

9242 

3o 

45 

2092 

9773 

0774 

4735 

9456 

9697 

8i38 

4659 

6820 

9622 

i5 

3o  o 

1962 

3.OCOO 

O622 

5ooo 

9282 

4.0000 

7942 

5ooo 

66o3 

S.oooo 

60   o 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

Course. 

I  40 


TRAVERSE     TABLE. 


Coarse. 

Dist.  I. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

Lut. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

o    / 

o    ^ 

3o  i5 

o8638 

o.5o38 

1.7377 

1.0075 

2.59l5 

I.5II3 

3.4553 

2.0l5l 

4-3i92 

2.5l89 

59  45 

3o 

8616 

5o75 

7233 

Ol5l 

5849 

5226 

4465 

0302 

3o8i 

5377 

3o 

45 

8594 

5n3 

7i88 

O226 

5782 

5339 

4376 

0452 

29-70 

5565 

i5 

Si   o 

8572 

5i5o 

7i43 

o3oi 

57i5 

545  1 

4287 

0602 

2858 

5752 

59   o 

i5 

8549 

5i88 

7098 

o375 

5647 

5563 

4i96 

o75i 

2746 

5939 

45 

3o 

8526 

5225 

7o53 

o45o 

5579 

5675 

4io6 

0900 

2632 

6i25 

3o 

45 

85o4 

5262 

7007 

o522! 

55n 

5786 

4oi^ 

1049 

25x8 

63ii 

i5 

32    0 

848o 

5299 

•  6961 

o598 

544i 

5898 

3922 

1197 

2402 

6496 

58   o 

i5 

8457 

5336 

6915 

o672 

5372 

6008 

3829 

1  345 

2286 

6681 

45 

3o 

8434 

5373 

6868 

o746 

53o2 

6119 

3736 

1492 

2I7O 

6865 

3o 

45 

o.84io 

o.54io 

1.6821 

1.0819 

2.523i 

1.6229 

3.3642 

2.i639 

4-2052 

2.7o49 

i5 

33   o 

8387 

5446 

6773 

0892 

5i6o 

6339 

3547 

1786 

1934 

7232 

57   o 

i5 

8363 

5483 

6726 

0966 

5089 

6449 

345i 

I932 

1814 

74i5 

45 

3o 

8339 

55i9 

6678 

1039 

5017 

6558 

3355 

2077 

1694 

7597 

3o 

45 

83i5 

5556 

6629 

mi 

4944 

6667 

3259 

2223 

i573 

7779 

ID 

34  o 

8290 

5592 

658i 

1184 

487i 

6776 

3i62 

2368 

i452 

796o 

56   o 

i5 

8266 

5628 

6532 

1256 

4798 

6884 

3o64 

25l2 

1329 

8i4o 

45 

3o 

8241 

5664 

6483 

:328 

4724 

6992 

2965 

2656 

1206 

8320 

3o 

45 

8216 

5700 

6433 

i4oo 

4649 

7100 

2866 

2800 

1082 

85oo 

i5 

35   o 

8192 

5736 

6383 

1472 

4575 

7207 

2766 

2943 

0958 

8679 

55   o 

i5 

0.8166 

0.5771 

1.6333 

i.i543 

2.4499 

i.73i4 

3.2666 

2.3o86 

4.o832 

2.8857 

45 

3o 

8i4i 

58o7 

6282 

1614 

4423 

742I 

2565 

3228 

o7o6 

9o35 

3o 

45 

8116 

5842 

623i 

i685 

4347 

7527 

2463 

337o 

o579 

92I2 

i5 

36   o 

8090 

5878 

6180 

i756 

4271 

7634 

236i 

35n 

o45i 

9389 

54   o 

i5 

8o64 

59i3 

6129 

1826 

4193 

7739 

2258 

3652 

O322 

9565 

45 

3o 

8039 

5948 

6077 

1896 

4n6 

7845 

2i54 

3793 

oi93 

974i 

3o 

45 

8oi3 

5983 

6o25 

1966 

4o38 

795o 

2o5o 

3933 

oo63 

99r6 

i5 

37   o 

7986 

6018 

5973 

2o36 

3959 

8o54 

1945 

4o73 

3.9932 

3.0091 

53   o 

i5 

7960 

6o53 

5920 

2106 

388o 

8i59 

i84o 

4212 

98oo 

0265 

45 

3o 

7934 

6088 

5867 

2175 

38oi 

8263 

i734 

435o 

9668 

o438 

3o 

45 

0.7907 

0.6122 

i.58i4 

1.2244 

2.372I 

1.8867 

3.1628 

2.4489 

3.9534 

3.o6n 

i5 

33   o 

7880 

6i57 

5760 

23i3 

364o 

847o 

l52O 

4626 

94os 

o783 

52    O 

i5 

7853 

6191 

5706 

2382 

356o 

8573 

i4i3 

4764 

9266 

o955 

45 

3o 

7826 

6225 

5652 

245o 

3478 

8675 

i3o4 

4901 

9i3o 

1126 

3o 

45 

7799 

6259 

5598 

25i8 

3397 

8778 

1195 

5o37 

8994 

I296 

i5 

39   o 

7771 

6293 

5543 

2586 

33i4 

8880 

1086 

5i73 

8857 

i466 

5i   o 

i5 

7744 

6327 

5488 

2654 

3232 

8981 

0976 

53o8 

8720 

i635 

45 

3o 

7716 

636i 

5432 

2722 

3i49 

9082 

o865 

5443 

858i 

1804 

3o 

45 

7688 

6394 

5377 

2789 

3o65 

9i83 

o754 

5578 

8442 

I972 

i5 

4o   o 

7660 

6428 

532i 

2856 

2981 

9284 

0642 

57I2 

83o2 

2i39 

5o   o 

15 

0.7632 

o.646i 

1.5265 

1.2922 

2.2897 

i.9384 

3.0529 

2  5845 

3.8162 

3.23o6 

45 

3o 

7604 

6494 

5208 

2989 

2812 

9483 

o4i6 

S978 

8020 

2472 

3o 

45 

7576 

6528 

5i5i 

3o55 

2727 

9583 

o3o3 

Oi  10 

7878 

2638 

i5 

4i  o 

7547 

656i 

5o94 

3l2I 

264l 

9682 

0188 

6242 

7735 

28o3 

49  o 

16 

75i8 

6593 

5o37 

3i87 

2555 

978o 

oo74 

6374 

7592 

2967 

45 

3o 

7490 

6626 

4979 

3252 

2469 

9879 

2.9958 

65o5 

7448 

3i3i 

3o 

45 

746i 

6659 

4921 

33i8 

2382 

9976 

9842 

6635 

73o3 

3294 

i5 

4a  o 

743  1 

6691 

'  4863 

3383 

2294 

2.0074 

9726 

6765 

7i57 

3457 

48   o 

i5 

7402 

6724 

48o4 

3447 

2207 

OI7I 

9609 

6895 

7on 

36i8 

45 

3o 

7373 

6756 

4746 

35i2 

2118 

0268 

9491 

7024 

6864 

378o 

3o 

45 

0.7343 

o.6788 

r.4686 

1.3576 

2.2O30 

2.o364 

3.9878 

2.7152 

3.67i6 

3.394o 

i5 

42  o 

73x4 

6820 

4627 

364o 

I94l 

o46o 

9254 

•7280 

6568 

4ioo 

47  o 

i5 

7284 

6852 

4567 

3704 

i85i 

o555 

9i35 

74o7 

64i9 

4259 

45 

3o 

7254 

6884 

45o7 

3767 

1761 

o65i 

9oi5 

7534 

6269 

44i8 

3o 

45 

7224 

69i5 

4447 

383o 

1671 

o745 

8895 

766i 

6118 

4576 

i5 

44  o 

7i93 

6947 

4387 

3893 

i58o 

o84o 

8774 

7786 

5967 

4733 

46  o 

i5 

7i63 

6978 

4326 

3956 

1489 

o934 

8652 

7912 

58i5 

489o 

45 

3o 

7i33 

•7009 

4265 

4oi8 

1898 

I027 

853o 

8o36 

5663 

5o45 

3o 

45 

7102 

7o4o 

4204 

4o8o 

i3o6 

II2O 

84o7 

8161 

55o9 

5201 

i5 

45  o 

7071 

7o7i 

4i42 

4i4s 

I2l3 

I2l3 

8284 

8284 

5355 

5355 

45   o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

i 

Dist.  1. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

bourse. 

TRAVERSE     IABLE. 


14 


Course. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lot. 

Dep. 

o    / 

*"~ 

"o    / 

3o  i5 

5.i83o 

3.0226 

6.o468 

3.5264 

6.9107 

4.0302 

7-7745 

4.534o 

8.6384 

5.o377 

59  45 

3o 

1698 

o452 

o3i4 

5528 

893o 

o6o3 

7547 

5678 

6i63 

o754 

3o 

45 

i564 

0678 

oi58 

579i 

8753 

o9o3 

7347 

6016 

594i 

1129 

i5 

3i   o 

i43o 

0902 

OOO2 

6o53 

8573 

1203 

7i45 

6353 

57i7 

i5o^ 

59   o 

i5 

I2Q5 

1126 

5.9844 

63i4 

839? 

1502 

6942 

6690 

549i 

1877 

45 

3o 

n58 

i35o 

9685 

6575 

8211 

1800 

6738 

7025 

5264 

225o 

3o 

45 

1021 

i573 

9525 

6835 

8028 

2097 

6532 

7359 

5o35 

2621 

i5 

32   0 

o883 

1795 

9363 

7094 

7844 

2394 

6324 

7693 

48o5 

2992 

58   o 

i5 

oy44 

2017 

9201 

7353 

7658 

2689 

6116 

8025 

4573 

336i 

45 

3o 

o6o3 

2238 

9o37 

7611 

7471 

2984 

59o5 

8357 

4339 

3730 

3o 

45 

5.o462 

3.2458 

5.8873 

3.7868 

6.7283 

4.3a78 

7.5694 

4.8688 

8,4io4 

5.4.097 

i5 

33   o 

O32O 

2678 

8707 

8i25 

7094 

357i 

548o 

9018 

3867 

4464 

57   o 

i5 

0177 

2898 

854o 

838i 

6903 

3863 

5266 

9346 

36a9 

4829 

45 

3o 

oo33 

3n6 

8372 

8636 

67n 

4i55 

5o5o 

9674 

3389 

5i94 

3o 

45 

4.9888 

3334 

8203 

8890 

65i8 

4446 

4832 

5.0001 

3i47 

5557 

i5 

34   o 

9742 

3552 

8o33 

9i44 

6323 

4735 

46i3 

0327 

2904 

59i9 

56   o 

i5 

9595 

3768 

7861 

9396 

6l27 

5o24 

4393 

o652 

2659 

6280 

45 

3o 

9448 

3984 

7689 

9648 

5930 

53i2 

4171 

0977 

a4i3 

664i 

3o 

45 

9299 

4200 

75i5 

99oo 

5732 

56oo 

3948 

i3oo 

2i65 

7000 

i5 

35   o 

9149 

44i5 

734i 

4-oi5o 

5532 

5886 

3724 

1622 

1915 

7358 

55   o 

i5 

4.8998 

3.4629 

5.7i65 

4«o4oo 

6.533i 

4-6i72 

7.3498 

5.i943 

8.i664 

5.77i5 

45 

3o 

8847 

4842 

6988 

o649 

5129 

6456 

3270 

2263 

l4l2 

8o7o 

3o 

45 

8694 

5o55 

6810 

o897 

4926 

674o 

3o42 

2582 

n57 

8425 

i5 

36   o 

854i 

5267 

663i 

n45 

472I 

7023 

2812 

2901 

0902 

8779 

54   o 

i5 

8387 

5479 

645  1 

l392 

45i6 

73o5 

258o 

32i8 

0644 

9i3i 

45 

3o 

823i 

5689 

6270 

i638 

43o9 

7586 

2347 

3534 

o386 

9482 

3o 

45 

8o75 

5899 

6088 

i883 

4ioo 

7866 

2Il3 

3849 

oia5 

9832 

i5 

37   o 

7918 

6109 

59o4 

2127 

3891 

8i45 

1877 

4i63 

7.9864 

6.0182 

53   o 

i5 

7760 

63i8 

5720 

237I 

368o 

8424 

i64o 

4476 

9600 

o529 

45 

3o 

7601 

6526 

5535 

26i3 

3468 

87oi 

1402 

4789 

9335 

o876 

3o 

45 

4.?44i 

3.6733 

5.5348 

4-2855 

6.3255 

4.8977 

7.1162 

5.5ioo 

7.9o69 

6.1222 

:5 

38   o 

7281 

6940 

5i6i 

3o96 

3o4i 

9253 

0921 

54io 

8801 

i5o6 

52    0 

i5 

7119 

7i46 

4972 

3337 

2826 

9528 

0679 

57i8 

8532 

I9o9 

45 

3o 

6956 

735i 

4?83 

3576 

2609 

98oi 

0435 

6026 

8261 

225l 

3o 

45 

6793 

7555 

4592 

38i5 

2391 

5.oo74 

0190 

6333 

7988 

2592 

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3o   o 

6629 

7759 

44oo 

4o52 

2I72 

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6.9943 

6639 

7715 

2932 

5i   o 

i5 

6464 

7962 

4207 

4289 

i95i 

0616 

9695 

6943 

7439 

327I 

45 

3o 

6297 

8i65 

4oi4 

4525 

1730 

0886 

9446 

7247 

7162 

36o8 

3o 

45 

6i3i 

8366 

38i9 

47^i 

i5o7 

n55 

9i96 

755o 

6884 

3944 

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4o  o 

5963 

8567 

3623 

4995 

1284 

1423 

8944 

785i 

66o4 

4279 

5o   o 

i5 

4.5794 

3.8767 

5.3426 

4-5229 

6.io59 

5.1690 

6.869i 

5.8i5i 

7.6323 

6.4612 

45 

3o 

5624 

8967 

3228 

546i 

o832 

i956 

8437 

845o 

6o4i 

4945 

3o 

45 

5454 

9166 

3o3o 

5693 

o6o5 

2221 

8181 

8748 

5756 

5276 

i5 

4i  o 

5283 

9364 

283o 

5924 

o377 

2485 

7924 

9045 

547i 

56o6 

49  o 

i5 

5no 

956i 

2629 

6i54 

oi47 

2748 

7666 

934i 

5i84 

5935 

45 

3o 

4937 

9757 

2427 

6383 

5.99i6 

3oio 

7406 

9636 

4896 

6262 

3o 

45 

4763 

9953 

2224 

6612 

9685 

327I 

7i45 

9929 

46o6 

6588 

i5 

4a  o 

4589 

4.oi48 

2O2O 

6839 

9452 

353o 

6883 

6.O222 

43i4 

69i3 

48   o 

i5 

44i3 

0342 

i8i5 

7066 

92i7 

3789 

6620 

o5i3 

4O22 

?237 

45 

3o 

4237 

o535 

i6o9 

7291 

8982 

4o47 

6355 

o8o3 

3728 

7559 

3o 

45 

44o59 

4.0728 

5.i4o3 

4.75i6 

5.8746 

5.43o4 

6.6o89 

6.IO92 

7.3432 

6.788o 

i5 

43  o 

388i 

O92O 

n95 

774o 

85o8 

456o 

5822 

i38o 

3i35 

8200 

47  o 

i5 

3702 

I  III 

0986 

7963 

82-70 

48i5 

5553 

1666 

2837 

85i8 

45 

3o 

3522 

i3oi 

0776 

8i85 

8o3o 

5o68 

5284 

I952 

2537 

8835 

3o 

45 

3342 

i49i 

o565 

84o6 

7789 

532i 

5oi3 

2236 

2236 

9i5i 

i5 

44  o 

3i6o 

1680 

o354 

8626 

7547 

5573 

474i 

25l9 

1934 

9466 

46   o 

i5 

2978 

1867 

oi4i 

8845 

73o4 

5823 

4467 

2801 

i63o 

9779 

45" 

3o 

2795 

2o55 

4.9928 

9o64 

7o6o 

6o73 

4i93 

3o82 

i325 

7.oo9i 

3o 

45 

2611 

2241 

97i3 

9281 

68i5 

632i 

39i7 

336i 

1019 

o4oi 

i5 

45  o 

2426 

2426 

9497 

9497 

6569 

6569 

364o 

364o 

O7II 

07I! 

45   o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist  9. 

Dist.  10. 

Course. 

I  42 


MERIDIONAL    PARTS. 


LATITUDE. 

Min. 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

Min. 

0 

o.o 

60.0 

120.  0 

180. 

240.2 

3oo.4 

360.7 

421. 

48i.6 

542.2 

6o3. 

664.1 

725.3 

0   , 

I 

1.0 

61.0 

21.0 

81. 

41.2 

01.4 

61.7 

22. 

82.6 

43.3 

04. 

65.i 

26.3 

I 

2 

2.O 

62.0 

22.0 

82. 

42.2 

02.4 

62.7 

23. 

83.6 

44.3 

o5. 

66.1 

27.4 

b 

3 

3.o 

63.o 

23.0 

83. 

43.2 

o3.4 

63.7 

24- 

84.6 

45.3 

06. 

67.1 

28.4 

3 

4 

4.0 

64-0 

24.0 

84. 

44.2 

o44 

64-7 

25. 

85.6 

46.3 

07. 

68.2 

29.^ 

.4 

e 

5.o 

65.o 

25.0 

85. 

45.2 

o5.4 

65.  7 

26. 

86.6 

47.3 

08.2 

69.2 

3o.5 

5 

6 

6.0 

66.0 

26.O 

86. 

46.2 

06.4 

66.7 

27. 

87.6 

48.3 

09.2 

70.2 

3i.5 

6 

7 

7.0 

67.0 

27.0 

87. 

47.2 

07.4 

67.7 

28. 

88.6 

49-3 

IO.2 

71.2 

32.5 

7 

8 

8.0 

68.0 

28.0 

88. 

48.2 

08.4 

68.7 

29. 

89.6 

5o.3 

II.  2 

72.2 

33.5 

8 

9 

9.0 

69.0 

29.0 

89. 

49.2 

09.4 

69.7 

3o, 

90.7 

5i.4 

12.2 

73.3 

34.5 

9 

10 

10.  0 

70.0 

iSo.o 

190. 

25O.2 

3io.4 

370.7 

43i. 

491.7 

552.4 

6l3.2 

674.3 

735.6 

10 

ii 

II.  0 

71.0 

3i.o 

91. 

5l.2 

11.4 

71.7 

32. 

92.7 

53.4 

14.2 

75.3 

36.6 

ii 

12 

12.  0 

72.0 

32.  0 

92. 

52.2 

12.4 

72.7 

33. 

93.7 

54-4 

i5.3 

76.3 

37.6 

it 

i3 

i3.o 

73.0 

33.o 

93. 

53.2 

i34 

73.7 

34-2 

94-7 

55.4 

i6.3 

77-3 

38.6 

i3 

i4 

i4-o 

74.0 

34.o 

94. 

54.2 

i44 

74.7 

35.2 

95.7 

56.4 

i7.3 

784 

39.6 

:4 

i5 

i5.o 

75.o 

35.o 

95. 

55.2 

i5.4 

75.7 

36.2 

0-6.7 

57-4 

i8.3 

79-4 

4o.7 

i5 

16 

16.0 

76.0 

36.o 

96. 

56.2 

16.4 

76.8 

37.2 

97-7 

58.4 

i9.3 

80.4 

4i,7 

16 

I? 

17.0 

77.0 

37.o 

97- 

57.2 

i7.5 

77.8 

38.2 

98.7 

59.4 

20.3 

81.4 

42.  7 

I7 

18 

18.0 

78.0 

38.o 

98. 

58.2 

1  8.5 

78.8 

39.2 

99.8 

6o.5 

21.3 

82.4 

43.7 

18 

T9 

19.0 

79.0 

39.0 

99. 

59.2 

i9.5 

79.8 

40.2 

5oo.8 

6i.5 

22.4 

83.5 

44.8 

'9 

20 

2O.O 

80.0 

i4o.o 

200. 

260.2 

320.5 

38o.8 

441-2 

5oi.8 

562.5 

623.4 

684.5 

745.8 

20 

21 

21.0 

81.0 

4i.o 

OI. 

6i.3 

21.5 

81.8 

42.2 

02.8 

63.5 

24.4 

85.5 

46.8 

21 

22 

22.0 

82.0 

42.0 

O2. 

62.3 

22.5 

82.8 

43.2 

o3.8 

64.5 

25.4 

86.5 

47-8 

22 

23 

23.0 

83.o 

43.o 

o3. 

63.3 

23.5 

83.8 

44-2 

o4-8 

65.5 

26.4 

87.5 

48.9 

23 

24 

24.0 

84-0 

44.o 

04. 

64-3 

24.5 

84.8 

45.2 

o5.8 

66.6 

27.4 

88.6 

49.9 

24 

25 

25.0 

85.o 

45.o 

o5. 

65.3 

25.5 

85.8 

46.3 

06.8 

67.6 

28.5 

89.6 

5o.9 

25 

26 

26.O 

86.0 

46.o 

06. 

66.3 

26.5 

86.8 

47.3 

07.8 

68.6 

29.5 

9o.6 

5i.9 

a  6 

27 

27.0 

87.0 

4?'.o 

07. 

67.3 

27.5 

87.8 

48.3 

08.9 

69.6 

3o.5 

9i.6 

53.o 

17 

28 

28.0 

88.0 

48.o 

08. 

68.3 

28.5 

88.8 

49-3 

09.9 

7o.6 

3i.5 

92.6 

54.o 

38 

29 

29.0 

89.0 

49.0 

09. 

69.3 

29.5 

89.8 

5o.3 

10.9 

7i.6 

32.5 

93.6 

55.o 

29 

3o 

3o.o 

90.0 

iSo.o 

210. 

270.3 

33o.5 

390.8 

45i.3 

511.9 

572.6 

633.5 

694.7 

756.0 

3o 

3i 

3i.o 

91.0 

5i.o 

II. 

7i.3 

3i.5 

91.8 

52.3 

12.9 

73.7 

34.6 

95.7 

57.i 

3i 

32 

32.0 

92.0 

52.  0 

12. 

72.3 

32.5 

92.9 

53.3 

i3.9 

74-7 

35.6 

96.7 

58.i 

32 

33 

33.o 

93.0 

53.i 

13. 

73.3 

33.5 

93.9 

54.3 

i4-9 

75.7 

36.6 

97-7 

59.i 

33 

34 

34.0 

94.0 

54. 

i4- 

74.3 

34.5 

94.9 

55.3 

i5.9 

76.7 

37.6 

98-7 

60.  i 

34 

35 

35.o 

gS.o 

55. 

i5. 

75.3 

35.5 

95.9 

56.3 

i6.9 

77-7 

38.6 

99.8 

61.1 

35 

36 

36.o 

96.0 

56. 

16. 

76.3 

36.5 

96.9 

57.3 

18.0 

78.7 

39.6 

7oo.8 

62.2 

36 

3? 

37.0 

97.0 

57. 

*7r 

77.3 

37.5 

97-9 

58.4 

19.0 

79-7 

40.7 

01.8 

63.2 

37 

38 

38.o 

98.0 

58. 

18.! 

78.3 

38.5 

98.9 

69.4 

20.  o 

80.8 

4i.7 

02.8 

64.2 

38 

39 

39.o 

99.0 

59. 

19.1 

79.3 

39.6 

99.9 

60.4 

2I.O 

81.8 

42.7 

o3.8 

65.2 

39 

4o 

4o.o 

IOO.O 

160. 

220.  2 

280.3 

34o.6 

400.9 

46i.4 

522.  0 

582.8 

643.7 

7o4-9 

766.3 

4o 

4i 

4i.o 

01.  0 

61. 

21.2 

8i.3 

4i.6 

01.9 

62.4 

23.0 

83.8 

44-7 

o5.9 

67.3 

4i 

42 

42.0 

02.  0 

62. 

22.2 

82.3 

42.6 

02.9 

63.4 

24.O 

84.8 

45.8 

06.  9 

68.3 

42 

43 

43.o 

o3.o 

63. 

23.2 

83.3 

43.6 

03.9 

644 

25.0 

85.8 

46.8 

°7-9 

69.3 

43 

44 

44-0 

o4-o 

64- 

24.2 

84.3 

44-6 

04.9 

65.4 

26.0 

86.8 

47-8 

o9.o 

7o.4 

44 

45 

45.o 

o5.o 

65. 

25.2 

85.3 

45.6 

o5.9 

66.4 

27. 

87.9 

48.8 

IO.O 

7i.4 

45 

46 

46.o 

06.0 

66. 

26.2 

86.3 

46.6 

07.0 

67.4 

28. 

88.9 

49-8 

II.  0 

72.4 

46 

4? 

47.0 

07.0 

67. 

27.2 

87.3 

47.6 

08.0 

68.4 

29. 

89.9 

5o.8 

12.0 

73.4 

47 

48 

48.o 

08.0 

68. 

28.2 

88.3 

48.6 

09.0 

69.5 

3o. 

9°-9 

5i.9 

i3. 

74.5 

48 

49 

49.0 

09.0 

69. 

29.2 

89.3 

49.6 

IO.O 

70.5 

3i. 

91.9 

52.9 

i4. 

75.5 

49 

5o 

5o.o 

IIO.O 

170. 

230.2 

290.3 

35o.6 

4n.o 

471.5 

532. 

592.9 

653.9 

7i5. 

776.5 

5o 

5i 

5i.o 

II.  0 

71- 

3l.2 

9i.3 

5i.6 

12.  0 

72.5 

33. 

93.9 

54.9 

16. 

77.5 

5i 

r 

52.0 

12.0 

72. 

32.2 

92.4 

52.6 

i3.o 

73.5 

34. 

95.0 

55.9 

17. 

78.6 

52 

53 

53.o 

i3.o 

73. 

33.2 

93.4 

53.6 

14.0 

74.5 

35. 

96.0 

57.0 

18.2 

79.6 

53 

54 

54.o 

14.0 

74- 

34.2 

944 

54.6 

i5.o 

75.5 

36.2 

97.0 

58.o 

19.2 

80.6 

54 

55 

55.o 

i5.o 

75. 

35.2 

95.4 

55.6 

16.0 

76.5 

37.2 

98.0 

59.o 

20.2 

81.7 

55 

56 

56.o 

16.0 

76. 

36.2 

96.4 

56.6 

17.0 

77.5 

38.2 

99.0 

60.0 

21.2 

82.7 

56 

57 

57.o 

17.0 

77- 

37.2 

97-4 

57.6 

18.0 

78.5 

39.2 

600.0 

61.0 

22.3 

83-7 

*>7 

58 

58.o 

18.0 

78.1 

38.2 

98.4 

58.6 

19.0 

79.5 

40.2 

01.  0 

62.1 

23.3 

84-7 

58 

59 

Sg.o 

19.0 

79.1 

39.2 

99-4 

59.7 

20.0 

8o.5 

41.2 

02.1 

63.i 

24.3 

85.8 

59 

MERIDIONAL    PARTS. 


145 


g 

LATITUDE. 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

3 

o 

786.8 

848.5 

910.5 

972.7 

io35.3 

1098.2 

n6i.5 

1225.  1 

1289.2 

i353.7 

i4i8.6 

i484-i 

o 

I 

87.8 

49-5 

n.5 

73.8 

36.3 

99.3 

62.5 

26.2 

90.3 

54-8 

19.7 

85.2 

i 

- 

88.8 

5o.5 

12.6 

74-8 

37.4 

iioo.S 

63.6 

27.3 

9i.3 

55.8 

20.8 

86.3 

2 

3 

89.9 

5i.6 

i3.6 

75.9 

38.4 

01.4 

64-7 

28.3 

92.4 

56.9 

21.9 

87.3 

3 

4 

90.9 

52.6 

i4-6 

76.9 

39.5 

02.4 

65.7 

29.4 

93.5 

58.o 

23-0 

88.4 

4 

5 

91.9 

53.6 

i5.7 

78.0 

4o.5 

o3.5 

66.8 

3o.4 

94.5 

59.o 

24.1 

89.5 

5 

6 

92.9 

54-7 

16.7 

79.0 

4i.6 

o4-5 

67.8 

3x.5 

95.6 

60.1 

25-1 

9o.6 

6 

7 

94.0 

55.7 

17.7 

80.0 

42.6 

o5.6 

68.9 

32.6 

96.7 

61.2 

26.2 

91.7 

7 

8 

gS.o 

56.7 

18.8 

81.1 

43.7 

06.6 

70.0 

33.6 

97.8 

62.3 

27.3 

92.8 

8 

9 

96.0 

57.8 

19.8 

82.1 

44-7 

07.7 

71.0 

34-7 

98.8 

63.4 

28.4 

93.9 

9 

10 

797.0 

858.8 

920.8 

983.2 

io45.8 

1108.7 

1172.1 

1235.8 

1299.9 

i364.5 

1429.5 

i495.o 

10 

ii 

98.1 

59.8 

21.9 

84.2 

46.8 

09.8 

73.i 

36.8 

iSoi.o 

65.6 

3o.6 

96.1 

ii 

12 

99.1 

60.9 

22.9 

85.2 

47-9 

10.8 

74.2 

37.9 

02.  o 

66.6 

3i.7 

97.2 

12 

i3 

800.1 

61.9 

23.9 

86.3 

48.9 

11.9 

75.2 

39.o 

o3.i 

67.7 

32.8 

98.3 

i3 

i4 

OI.2 

62.9 

25.  0 

87.3 

49.9 

12.9 

76.3 

4o.o 

o4-2 

68.8 

33.9 

99.4 

i4 

i5 

02.2 

64.o 

26.0 

88.4 

5i.o 

14.0 

77-4 

4i.i 

o5.3 

69-9 

34.9 

i5oo.5 

i5 

16 

03.2 

65.o 

27.0 

89.4 

52.  0 

i5.o 

78.4 

42.2 

o6.3 

70.0 

36.o 

01.6 

16 

!? 

04.2 

66.0 

28.1 

90.4 

53,i 

16.1 

79.5 

43.2 

07.4 

72.0 

37.i 

02.7 

17 

18 

o5.3 

67.1 

29.1 

9i.5 

54-1 

17.1 

8o.5 

44.3 

o8.5 

73.i 

38.2 

o3.8 

18 

J9 

o6.3 

68.1 

3o.i 

92.5 

55.2 

18.2 

81.6 

45.4 

09.6 

74.2 

Sg.S 

04.9 

:9 

20 

807.3 

869.1 

931.2 

993.6 

io56.2 

1119.2 

1182.7 

1248.4 

i3io.6 

i375.3 

i44o.4 

i5o6.o 

20 

21 

08.4 

70.1 

32.2 

94-6 

57.3 

20.3 

83.7 

47-5 

11.7 

76.4 

4i.5 

07.1 

21 

22 

09.4 

71.2 

33.3 

95.6 

58.3 

21.3 

84-8 

48.6 

12.8 

77-4 

42.6 

08.2 

22 

23 

10.4 

72.2 

34.3 

96-7 

59.4 

22.4 

85.8 

49-6 

i3.8 

78.5 

43.7 

o9.3 

23 

24 

11.4 

73.2 

35.3 

97-7 

60.4 

23.4 

86.9 

50.7 

i4.9 

79.6 

44-8 

10.4 

24 

25 

12.5 

74.3 

36.3 

98.8 

61.4 

24.5 

88.0 

5i.8 

16.0 

80.7 

45.8 

ii.5 

25 

26 

i3.5 

75.3 

37.4 

99-8 

62.5 

25.5 

89.0 

52.8 

17.1 

81.8 

46.9 

12.6 

26 

27 

i4.5 

76.3 

38.4 

1000.8 

63.5 

26.6 

90.1 

53.9 

18.1 

82.8 

48.o 

i3.7 

27 

28 

i5.5 

77-4 

39.5 

01.  9 

64.6 

27.6 

91.1 

55.o 

19.2 

83.9 

49.1 

i4.8 

28 

29 

1  6.6 

78.4 

4o.5 

02.9 

65.6 

28.7 

92.2 

56.o 

20.3 

85.o 

50.2 

i5.9 

29 

3o 

817.6 

8794 

94i.6 

i  oo4.o 

io66.7 

1129.7 

II93.2 

1257.1 

i32i.4 

!386.i 

i45i.3 

1517.0 

3o 

3i 

18.6 

8o.5 

42.6 

o5.o 

67.7 

3o.8 

94.3 

58.2 

22.5 

87.2 

52.4 

18.1 

3i 

32 

19.6 

8i.5 

43.6 

06.  i 

68.8 

3i.8 

954 

59.2 

23.5 

88.3 

53.5 

19.2 

32 

33 

20.7 

82.5 

44-7 

07.1 

69.8 

32.9 

964 

60.3 

24.6 

89.^ 

54.6 

20.  3 

33 

34 

21.7 

83.6 

45.7 

08.  i 

70.9 

34.o 

97.5 

61.4 

25.7 

90.^ 

55.6 

21.2 

34 

35 

22.7 

84-6 

46.7 

09.2 

72.0 

35.i 

98.5 

62.4 

26.7 

9i.5 

56.7 

22.5 

35 

36 

23.8 

85.6 

47.8 

IO.2 

73.o 

36.i 

99.6 

63.5 

27.8 

92.6 

57.8 

23.6 

36 

37 

24.8 

86.7 

48.8 

ii.  3 

74.1 

37.2 

I2OO.7 

64.6 

28.9. 

93.7 

58.9 

24.7 

37 

38 

25.8 

87.7 

49.9 

12.3 

75.i 

38.2 

01.7 

65.6 

3o.o 

94.8 

60.0 

25.8 

38 

39 

26.9 

88.7 

5o.9 

i3.4 

76.2 

39.3 

02.  8 

66.7 

3x.x 

95.8 

61.1 

26.9 

39 

4o 

827.9 

889.8 

oSi.g 

ioi44 

1077.2 

n4o.3 

1203.9 

1267.8 

i332.i 

1396.9 

1462.2 

i528.o 

4o 

4i 

28.9 

90.8 

53.o 

i5.4 

78.3 

4i.4 

o4-9 

68.8 

33.2 

98.0 

63.3 

29.I 

4i 

42 

29.9 

91.8 

54-0 

i6.5 

79.3 

42.4 

06.0 

69.9 

34.3 

99.1 

64-4 

30.2 

42 

43 

3i.o 

92.9 

55.i 

i7.5 

80.4 

43.5 

07.1 

71.0 

35.3 

I4OO.2 

65.5 

3i.3 

43 

44 

32.0 

93.9 

56.i 

18.6 

81.4 

44.6 

08.  i 

72.1 

36.4 

01.  C 

66.6 

32.4 

44 

45 

33.o 

94.9 

57.! 

19.6 

82.5 

45.6 

09.2 

73.i 

37.5 

02.  /. 

67.7 

33.5 

45 

46 

34-1 

96.0 

58.2 

20.6 

83.5 

46.7 

•    10.2 

74.2 

38.6 

o3.4 

68.8 

34.6 

46 

47 

35.i 

97.0 

59.2 

21.7 

84-6 

47-7 

n,3 

75.3 

39.7 

o4.5 

69.8 

35.7 

47 

48 

36.i 

98.0 

60.2 

22.7 

85.6 

48.8 

12.4 

76.3 

4o.7 

o5.6 

7°-9 

36.8 

48 

49 

37.2 

99.1 

6i.3 

23.8 

86.7 

49.9 

i3.4 

77-4 

4i.8 

06.7 

72.0 

37.9 

49 

5o 

838.2 

900.1 

962.3 

1024.8 

1087.7 

n5o.9 

I2i4.5 

i278.5 

i342.9 

1407.8 

i473.x 

iSSg.o 

5o 

5i 

39.2 

01.  1 

63.4 

25.9 

88.8 

52.0 

i5.5 

79.5 

44.o 

08.8 

74.2 

4o.i 

5i 

52 

4O.2 

O2.  2 

644 

26.9 

89.8 

53.o 

16.6 

80.6 

45.i 

09.9 

75.3 

41.2 

52 

53 

4i.3 

03.2 

65.5 

28.0 

9°  9 

54.i 

17.7 

81.7 

46.i 

II.  0 

76.4 

42.3 

53 

54 

42.3 

o4.3 

66.5 

29.0 

91.9 

55.i 

18.7 

82.8 

47.2 

12.  1 

77.5 

43.4 

54 

55 

43.3 

o5.3 

67.5 

3o.i 

93.o 

56.2 

19.8 

83.8 

48.3 

1  3.2 

78.6 

44-5 

55 

56 

444 

o6.3 

68.6 

3i.i 

.   94-0 

57.2 

20.9 

84-9 

49.4 

i4.3 

79-7 

45.6 

56 

57 

45.4 

07.4 

69.6 

32.2 

9§4 

58.3 

21.9 

86.0 

5o.4 

i5.4 

80.8      46.7 

57 

58 

46.4 

08.; 

70.7 

33.2 

96.1 

59.4 

23.0 

87.0 

5i.5 

1  6.5 

8i.9!     47-8 

58 

59 

47-5 

09.4 

71.7 

34-3 

97.2 

60.4 

24.1 

88.1 

52.6 

t 

i7.5 

83.o      48.9 

59 

14* 


MERIDIONAL    FARTS. 


LATITUDE. 

Mia. 

25° 

26° 

27° 

28° 

29° 

30° 

31° 

32° 

33° 

34° 

35° 

Mia. 

o 

i55o.o 

i6i6.5 

1683.5 

1751.2 

i8i9.4 

1888.4 

i958.o 

2028.4 

2099.5 

2171.5 

2244-3 

O 

i 

5i.i 

17.6 

84-6 

52.3 

20.6 

89.5 

59.2 

29.6 

2100.7 

72.7 

45.5 

I 

2 

52.2 

18.7 

85.8 

53.4 

21.7 

90.7 

60.4 

3o.7 

oi.9 

73.9 

46.8 

2 

3 

53.3 

19.8 

86.9 

54-6 

22.  9 

91.9 

61.6 

3i.9 

o3.i 

75.i 

48.o 

3 

4 

54-4 

20.9 

88.0 

55.7 

24.O 

93.0 

62.7 

33.i 

o4.3 

76.3 

49.2 

4 

5 

55.5 

22.  0 

89.1 

56.8 

25.2 

94.1 

63.9 

34.3 

o5.5 

77.5 

5o4 

5 

6 

56.6 

23.a 

90.3 

58.o 

26.3 

95.3 

65.o 

35.5 

06.7 

78.7 

5i.6 

6 

7 

57.7 

24.3 

91.4 

59.i 

27.5 

96.5 

66.2 

36.7 

°7-9 

80.0 

52.9 

7 

8 

58.8 

25.4 

92.5 

60.2 

28.6 

97.6 

67.4 

37.8 

o9.i 

81.2 

54-1 

8 

9 

59.9 

26.5 

93.6 

61.4 

29-7 

98.8 

68.5 

39.0 

io.3 

82.4 

55.3 

9 

13 

i56i.o 

1627.6 

1694.8 

1762.5 

i83o.9 

i899.9 

i969.7 

2O4O.2 

21  I  1.5 

ai83.6 

2256.5 

10 

II 

62.1 

28.7 

95.9 

63.6 

32.0 

I9OI.I 

7°-9 

4i4 

12.7 

84.8 

57.8 

i: 

12 

63.2 

29.8 

97.0 

64.8 

33.2 

02.3 

72.0 

42.6 

i3.9 

86.0 

Sg.o 

12 

i3 

644 

3i.o 

98.1 

65.9 

34-3 

o3.4 

73.2 

43.8 

i5.i 

87.2 

60.2 

i3 

r4 

65.5 

32.1 

99.3 

67.0 

35.5 

o4.6 

74-4 

44-9 

i6.3 

884 

61.4 

i4 

i5 

66.6 

33.2 

1700.4 

68.2 

36.6 

o5.7 

75.6 

46.i 

i7.5 

89.6 

62.  7 

16 

16 

67.7 

34.3 

oi.5 

69.3 

37.8 

06.  9 

76.8 

47.3 

18.7 

90.8 

63.9 

16 

I? 

68.8 

35.4 

02.  6 

70.5 

38.9 

08.1 

77-9 

48.5 

19.8 

92.0 

65.i 

17 

18 

69.9 

36.5 

o3.8 

71.6 

4o.i 

09.2 

79.i 

49-7 

21.  0 

93.3 

66.3 

18 

J9 

71.0 

37.6 

04.9 

72.7 

4l>2 

10.4 

80.2 

5o.8 

.23.  2 

944 

67.5 

19 

20 

1572.1 

1638.8 

1706.0 

1773.9 

1842.4 

1911.5 

i98i.4 

2052.0 

21234 

2i95.7 

2268.8 

20 

21 

73.2 

39.9 

07.1 

75.0 

43.5 

12.  7 

82.6 

53.2 

24.6 

96.9 

70.0 

21 

22 

74-3 

4i.o 

o8.3 

76.1 

44.6 

i3.8 

83.7 

544 

25.8 

98.1 

71.2 

22 

23 

754 

42.1 

09.4 

77.2 

45.8 

i5.o 

84.9 

55.6 

27.0 

99.3 

72.5 

23 

24 

76.5 

43.2 

10.5 

78.4 

46.9 

16.2 

86.1 

56.8 

28.2 

22OO.5 

73.7 

24 

25 

77.6 

44-3 

ii.  6 

79.5 

48.i 

I7.3 

87.3 

58.o 

29.4 

01.7 

74-9 

25 

26 

78.7 

45.5 

12.8 

80.6 

49.2 

i8.5 

884 

59.i 

3o.6 

o3.o 

76.1 

26 

27 

79.8 

46.6 

i3.9 

81.8 

5o.4 

19.6 

89.6 

6o.3 

3i.8 

04.2 

774 

27 

28 

80.9 

47-7 

i5.o 

83.o 

5i.5 

20.8 

9o.8 

6i.5 

33.o 

o54 

78.6 

28 

29 

82.1 

48.8 

16.1 

84-i 

52.7 

21.  9 

92.0 

62.7 

34.2 

06.6 

79.8 

29 

3o 

i583.a 

1649-9 

1717.3 

1785.2 

i853.8 

I923.I 

I993.i 

2063.9 

2i354 

2207.8 

2281.0 

3o 

3i 

84.3 

$1.0 

18.4 

86.4 

55.o 

24-3 

94-3 

65.i 

36.6 

09.0 

82.3 

3i 

32 

85.4 

52.2 

19.5 

87.5 

56.i 

25.4 

95.5 

66.2 

37.8 

10.2 

83.5 

32 

33 

86.5 

53.3 

20.7 

88.6 

57.2 

26.6 

96.6 

67.4 

39.o 

11.4 

84-7 

33 

34 

87.6 

544 

21.8 

89.8 

58.4 

27.8 

97.8 

68.6 

4O.2 

12.7 

86.0 

34 

35 

88.7 

55.5 

22.9 

90.9 

59.6 

28.9 

99.0 

69.8 

4i4 

13.9 

87.2 

35 

36 

89.8 

56.6 

24.0 

92.1 

60.7 

3o.i 

2OOO.2 

7i.o 

42.6 

i5.i 

884 

36 

37 

90.9 

57.8 

25.2 

93.2 

61.9 

3i.3 

01.3 

72.2 

43.8 

i6.3 

89.7 

37 

38 

92.0 

58.9 

26.3 

94-3 

63.o 

32.4 

02.  5 

734 

45.o 

i7.5 

9°-9 

38 

39 

$3.z 

60.0 

274 

95.5 

64.2 

33.6 

o3.7 

74.5 

46.2 

18.7 

92.1 

39 

4o 

i594-3 

1661.1 

1728.6 

1796.6 

i865.3 

1934.7 

2004.9 

2075.7 

2147.4 

2219.9 

2293.3 

4o 

4i 

95.4 

62.2 

29.7 

97.8 

66.5 

35.9 

06.0 

76.9 

48.6 

21.2 

94.6 

4i 

42 

96.5 

63.4 

3o.8 

98.9 

67.6 

37.i 

O7.2 

78.1 

49.8 

22.4 

95.8 

42 

43 

97.6 

64.5 

81.9 

1800.0 

68.8 

38.2 

084 

79.3 

5i.o 

23.6 

97.o 

43 

44 

98.7 

65.6 

33.i 

OI.2 

69.9 

39.4 

09.6 

80.5 

52.2 

24.8 

98.3 

44 

45 

99.8 

66.7 

34.2 

02.3 

71.1 

4o.5 

I0.7 

81.7 

534 

26.0 

99.5 

45 

46 

1600.9 

67.8 

35.3 

o3.5 

72.2 

4i.7 

II.9 

82.9 

54.6 

27.2 

23oo.7 

46 

47 

02.  o 

69.0 

36.5 

o4.6 

73.4 

42.9 

i3.i 

84.o 

55.8 

28.5 

02.0 

47 

48 

63.1 

70.1 

37.6 

o5.7 

74-5 

44.o 

i4-3 

85.2 

57.0 

29.7 

03.2 

48 

49 

o4-2 

71.2 

38.7 

06.9 

75.7 

45.2 

1  54 

864 

58.2 

30.9 

o44 

49 

5o 

i6o5.4 

1672.3 

1739.9 

1808.  o 

1876.8 

i946.4 

2016.6 

2087.6 

2159.4 

2232.1 

23o5.7 

5o 

5i 

o6.5 

73.4 

4i.o 

09.2 

78.0 

47.5 

17.8 

88.8 

60.7 

33.3 

o6.9 

5i 

52 

07.6 

74.5 

42.1 

io.3 

79.2 

48.7 

19.0 

90.0 

61.9 

34-6 

08.  1 

52 

53 

08.7 

75.7 

43.2 

11.4 

8o.3 

49.9 

20.2 

91.2 

63.i 

35.8 

o94 

53 

54 

09.8 

76.8 

444 

12.6 

8i.5 

5i.o 

21.3 

92.4 

64-3 

37.o 

10.6 

54 

55 

10.9 

77.9 

45.5 

i3.7 

82.6 

52.2 

22.5 

93.6 

65.5 

38.2 

ix.8 

55 

56 

12.0 

79.0 

46.6 

i4.9 

83.8 

53.4 

23.7 

94.8 

66.7 

394 

i3.i 

56 

57 

i3.i 

/  y 

80.2 

47-8 

16.0 

84-9 

54.5 

24.9 

96.0 

67.9 

4o.7 

i4.3 

57 

58 

14.2 

8i.3 

48.9 

I7.2 

86.1 

55.7 

26.O 

97.1 

69.1 

4i-9 

i5.5 

58 

59 

1  5,4 

82.4 

5o.o 

i8.3 

87.2 

56.9 

27.2 

98.3 

70.3 

43.i 

16.8 

59 

MERIDIONALPARTS.                                     M; 

1 
LATITUDE. 

Min 

36° 

37° 

38° 

39° 

40° 

41° 

42° 

43° 

44° 

45° 

46° 

Min. 

o 

23i8.o 

2392.6 

2468.3 

2545.0 

2622.7 

2701.6 

2781.7 

2863.1 

2945.8 

3o3o.o 

3n5.6 

O 

I 

19.2 

93.9 

69.5 

46.2 

24.0 

02.9 

83.i 

64-5 

47-2 

3i4 

17.0 

I 

2 

20.5 

95.i 

70.8 

47-5 

25.3 

04.3 

844 

65.8 

48.6 

32.8 

i8.5 

2 

3 

21.7 

96.4 

72.1 

48.8 

26.6 

o5.6 

85.8 

67.2 

5o.o 

34-2 

i9.9 

3 

4 

23  0 

97-7 

73.4 

5o  [ 

27.9 

06.9 

87.1 

68.5 

5i4 

35.6 

aij 

4 

5 

242 

98.9 

74.6 

5i.4 

29.2 

o8.3 

88.5 

70.0 

52.8 

37.0 

22.8 

5 

6 

25.4 

2400.2 

75.9 

52.7 

3o.5 

09.6 

89.8 

7i.3 

54.2 

384 

24-2 

6 

7 

26.7 

01.4 

77.1 

54.0 

3i.9 

10.9 

91.2 

72.7 

55.6 

39.8 

25.7 

7 

8 

27.9 

02.7 

78.5 

55.3 

33.2 

12.2 

92.5 

74.i 

57.o 

4i.3 

27.I 

8 

9 

29.1 

o3.9 

79-7 

56.6 

34.5 

i3.5 

93.8 

754 

58.3 

42.7 

28.5 

9 

10 

233o.4 

24o5.2 

2481.0 

2557.8 

2635.8 

2714.9 

2795.i 

2876.8 

2959.8 

3o44.i 

3i3o.o 

10 

ii 

3i.6 

06.4 

82.2 

59.i 

37.i 

16.2 

96.5 

78.2 

61.1 

45.5 

3i.5 

TI 

12 

32.9 

07.7 

83.5 

60.4 

38.4 

17.5 

97-9 

79.5 

62.5 

47-0 

32.9 

12 

i3 

34-1 

09.0 

84.8 

61.7 

39.7 

18.9 

99-3 

8o.9 

63.9 

484 

34.3 

13 

i4 

35.3 

1O.2 

86.1 

63.o 

4i.o 

2O.2 

2800.6 

82.3 

65.3 

49.8 

35.8 

14 

i5 

36.6 

n.5 

87.4 

64-3 

42.3 

21.5 

02.  o 

o3.7 

66.7 

5l.2 

37.2 

i5 

16 

37.8 

12.7 

88.6 

65.6 

43.6 

22.9 

o3.3 

85.o 

68.1 

52.6 

38.7 

16 

i? 

39.0 

14.0 

89.9 

66.9 

44-9 

24.2 

04.7 

864 

69.5 

54.i 

4o.i 

*7 

18 

4o.3 

15.2 

91.2 

68.2 

46.3 

25.5 

06.0 

87.8 

•70.9 

55.5 

4i.6 

18 

19. 

4i.5 

i6.5 

92.4 

69.5 

47-6 

26.8 

07.3 

89.i 

72.3 

56.9 

43.o 

19 

20 

2342.8 

2417-8 

2493.7 

2570.7 

2648.9 

2728.2 

2808.8 

289o.5 

2973.7 

3o58.3 

3i44-5 

20 

21 

44.o 

19.0 

95.0 

72.0 

50.2 

29.5 

IO.I 

91.9 

76.1 

59.7 

45.9 

21 

22 

45.3 

20.3 

96.3 

73.3 

5i.5 

3o.8 

11.4 

93.3 

76.5 

61.2 

474 

22 

23 

46.5 

21.5 

97.6 

74-6 

52.8 

32.2 

12.8 

94.7 

77-9 

62.6 

48.8 

23 

24 

47-7 

22.8 

98.8 

75.9 

54.i 

33.5 

14.1 

96.0 

79-3 

64.0 

5o.3 

24 

25 

49.0 

24.0 

2500.1 

77.2 

55.5 

34.8 

i5.5 

974 

8o.7 

654 

5i.7 

25 

26 

5o.2 

25.3 

01.4 

78.5 

56.8 

36.2 

16.8 

98.8 

82.1 

66.9 

53.2 

26 

27 

5i.5 

26.5 

02.7 

79.8 

58.i 

37.5 

18.2 

2900.2 

83.5 

68.3 

54.6 

27 

28 

52.7 

27.8 

o3.9 

81.1 

59.4 

38.8 

I9.5 

oi.5 

84.9 

69-7 

56.i 

28 

29 

54.o 

29.1 

O5.2 

82.4 

60.7 

4O.2 

2O.9 

02.9 

86.3 

71.1 

57.5 

29 

3o 

2355.2 

243o.3 

25o6.5 

2583.7 

2662.0 

274l.5 

2822.3 

2904.3 

2987.7 

3072.6 

3i59.o 

3o 

3i 

56.5 

3i.6 

07.8 

85.o 

63.3 

42.9 

23.6 

o5-7 

89.1 

74.o 

60.4 

3i 

32 

57.7 

32.9 

09.0 

86.3 

64.6 

44-2 

25.0 

07.1 

90.5 

754 

61.9 

32 

33 

58-9 

34.1 

io.3 

87.6 

66.0 

45.5 

26.3 

08.4 

91.9 

76.9 

63.3 

33 

34 

60.2 

35.4 

u.6 

88.9 

67.3 

46.9 

27.7 

09.7 

93.3 

78.3 

64.8 

34 

35 

61.4 

36-7 

12.9 

90.2 

68.6 

48.2 

29.0 

II.  2 

94.7 

79-7 

66.2 

35 

36 

62.7 

37.9 

14.2 

9i.5 

69.9 

49.5 

3o4 

12.6 

96.i 

81.1 

67.7 

36 

37 

63.9 

39.2 

i5.4 

92.8 

71.2 

5o.9 

3i.7 

14.0 

97-5 

82.6 

69.i 

37 

38 

65.2 

4o.4 

16.7 

94.1 

72.5 

52.2 

33.i 

i5.3 

98.9 

84-0 

7o.6 

38 

39 

66.4 

4i-7 

18.0 

954 

73.9 

53.5 

34-5 

16.7 

3ooo.3 

854 

72.O 

39 

4o 

2367.6 

2443.o 

25i9.3 

2596.7 

2675.2 

2754.9 

2835.8 

2918.1 

3ooi.8 

3o86.9 

3i73.5 

4o 

4i 

68.9 

44.2 

20.5 

98.0 

76.5 

56.2 

37.2 

i9.5 

03.2 

88.3 

75.o 

4i 

42 

70.2 

45.5 

21.8 

99.3 

77.8 

57.6 

38.6 

2O.9 

o4-6 

89-7 

764 

42 

43 

71.4 

46.8 

23.1 

2600.5 

79.1 

58.9 

39.9 

22.3 

06.0 

91.2 

77-9 

43 

44 

72.6 

48.o 

24.4 

01.9 

8o.5 

60.2 

4i.3 

23.6 

07.4 

92.6 

79-3 

44 

45 

73.9 

49.3 

25.7 

03.2 

81.8 

6i.5 

42.6 

25.  0 

08.8 

94.o 

80.8 

45 

46 

75.i 

5o.6 

27.0 

o4.5 

83.i 

62.9 

44.o 

26.4 

10.2 

95.5 

82.3 

46 

47 

76.4 

5i.8 

28.3 

o5.8 

844 

64.3 

454 

27.8 

11.6 

96.9 

83.7 

47 

48 

77.6 

53.i 

29.5 

07.1 

85.7 

65.6 

46.7 

29.2 

i3.o 

98.3 

85.2 

48 

49 

78.9 

54-3 

3o.8 

08.4 

87.1 

66.9 

48.i 

3o.6 

i44 

99-7 

86.6 

49 

5o 

s38o.i 

2455.6 

2532.1 

2609.7 

2688.4 

2768.3 

2849.5 

2932.O 

3oi5.8 

SlOI.2 

3i88.i 

5o 

5i 

81.4 

56.9 

33.4 

II.O 

89.7 

69.6 

5o.8 

33.3 

17.2 

02.6 

89.6 

5i 

52 

82.6 

58.i 

34-7 

12.3 

91.0 

71.0 

52.2 

34-7 

18.7 

04.1 

9i.o 

52 

53 

83.9 

59.4 

36.o 

1  3.6 

92.3 

72.3 

53.5 

36.i 

20.  i 

o5.6 

92.5 

53 

54 

85.i 

60.7 

37.2 

i4-9 

93.7 

73.7 

54.9 

37.5 

21.5 

07.0 

94-o 

54 

55 

86.4 

61.9 

38.5 

16.2 

95.0 

75.o 

56.3 

38.9 

22.  9 

08.4 

954 

55 

56 

87.6 

63.2 

39.8" 

1  17.5 

96.3 

76.3 

57.7 

4o.3 

24-3 

o9.8 

96.9 

56 

57 

88.9 

64.5 

4i.i 

18.8 

97.6 

77-7 

59.o 

41.7 

25.7 

II.  2 

98.4 

57 

58 

90.2 

65.8 

42.4 

2O.I 

99.0 

79-° 

6o.5 

43.i 

27.1 

12.7 

09  8|  58 

59 

91.4 

67.0 

43.6 

21.4 

2700.3 

80.4 

6T.T 

444 

28.5 

i4«i  32OI.3    59 

146 


MERIDIONAL    PARTS. 


LATITUDE. 

Min. 

47° 

48° 

49° 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

Min. 

O 

8202.7 

3291.5 

3382.1 

3474-5 

3568.8 

3665.2 

3763.8 

3864.6 

3968.0 

4073.9 

4182.6 

o 

I 

04.2 

93.0 

83.6 

76.0 

7o.4 

66.8 

65.4 

66.3 

69.7 

75.7 

84.5 

I 

2 

o5.7 

94-5 

85.i 

77.6 

72.0 

b8.4 

67.1 

68.0 

7i.5 

77.5 

86.3 

2 

3 

07.1 

96.0 

86.7 

79.1 

73.6 

7O.I 

68.8 

69,7 

73.2 

79-3 

88.1 

3 

4 

08.6 

97.5 

88.2 

80.7 

75.2 

71.7 

70.4 

7i.5 

75.0 

61.1 

90.0 

4 

5 

I  O.I 

99.0 

89.7 

82.3 

76.8 

73.3 

72.1 

73.2 

76.7 

82.9 

91.8 

5 

6 

ii.  5 

33oo.5 

91.3 

83.8 

78.4 

75.o 

73-7 

74-9 

78.4 

84.7 

93-7 

6 

7 

i3.o 

02.  o 

92.8 

85.4 

79-9 

76.6 

75.4 

76.6 

80.2 

86.4 

95.5 

7 

8 

i4-5 

o3.5 

94-3 

87.0 

8i.5 

78.2 

77.1 

78.3 

82.0 

88.2 

97-3 

8 

9 

!5.9 

o5.o 

95.8 

88.5 

83.i 

79.8 

78.7 

80.0 

83.7 

90.0 

99.2 

9 

10 

3217.4 

33o6.5 

3397.4 

3490.1 

3584-7 

368i.5 

378o.4 

388i.7 

3985.4 

4091.8 

4201.0 

10 

ii 

18.9 

08.0 

98.9 

91.6 

86.3 

83.i 

82.1 

83.4 

87.2 

93.6 

02.9 

ii 

12 

20.3 

o9.5 

34oo.4 

93.2 

87.9 

84-7 

83.7 

85..I 

88.9 

95.4 

o4-7 

12 

i3 

21.8 

II.  0 

02.0 

94-7 

89.5 

86.4 

85.4 

86.8 

90.7 

97.2 

06.6 

i3 

i4 

23.3 

12.5 

o3.5 

96.3 

91.1 

88.0 

87.1 

88.5 

92.5 

99.0 

08.4 

U 

i5 

24.8 

i4«o 

o5.o 

97-9 

92.7 

89.6 

88.8 

90.2 

94.2 

4ioo.8 

io.3 

i5 

16 

26.2 

i5.5 

06.6 

99.4 

94-3 

9i.3 

90.4 

92.0 

96.0 

02.6 

12.  1 

16 

I7 

27.7 

17.0 

08.  i 

35oi.o 

95.9 

92.9 

92.1 

93-7 

97-7 

o4-4 

i4«o 

J7 

18 

29.2 

i8.5 

09.6 

02.6 

97.5 

94-5 

93.8 

95.4 

99.5 

06.2 

i5.8 

18 

19 

3o.7 

2O.O 

ii.  i 

o4«i 

99.1 

96.2 

95.5 

97-i 

iOQI.2 

08.0 

17.7 

J9 

20 

3232.1 

3321.5 

3412.7 

35o5.7 

^600.7 

3697.8 

3797-1 

389£.8 

iooS.o 

4109.8 

42i9.5 

20 

21 

33.6 

23.0 

14.2 

o7.3 

02.3 

99.4 

98.8 

3900.5 

04.7 

n.6 

21.4 

21 

22 

35.i 

24-5 

1.5.7 

08.8 

03.9 

37oi.i 

38oo.5 

O2.  2 

06.5 

1  3.4 

23.2 

22 

23 

36.6 

26.0 

17.3 

10.4 

o5.5 

02.  7 

O2.  2 

o4«o 

08.3 

15.2 

25-1 

23 

24 

38.o 

27.5 

18.8 

12.0 

07.1 

o44 

o3.8 

o5.7 

IO.O 

17.1 

27.O 

24 

25 

39.5 

29.0 

20.4 

i3.5 

08.7 

06.0 

o5.5 

07.4 

n.8 

18.9 

28.8 

25 

26 

4i.o 

3o.6 

21.9 

i5.i 

io.3 

07.6 

07.2 

09.1 

i3.5 

20.7 

3o.6 

26 

27 

42.5 

32.1 

23.5 

16.7 

11.9 

09.3 

08.9 

10.8 

i5.3 

22.5 

32.5 

27 

28 

44.o 

33.6 

25.  0 

i8.3 

i3.6 

I0.§ 

io.5 

12.5 

17.1 

24.3 

34.4 

28 

29 

45.4 

35.i 

26.5 

19.8 

i5.i 

12.6 

12.2 

i4.3 

18.8 

26.1 

36.2 

29 

3o 

3246.9 

3336.6 

3428.0 

3521.4 

3616.7 

3714.2 

38i3.9 

3916.0 

[O2O.6 

4127.9 

4238.1 

3o 

3i 

48.4 

38.i 

29.6 

23.0 

18.4 

16.8 

1  5.6 

17-7 

22.4 

29.7 

4o.o 

3i 

32 

49.9 

39.6 

3i.i 

24.6 

20.  o 

i7.5 

i7.3 

19.4 

24.1 

3i.5 

4i.8 

32 

33 

5i.4 

4i.i 

32.7 

26.1 

21.6 

19.1 

18.9 

21.2 

25.9 

33.3 

43.7 

33 

34 

52.8 

42.6 

34.2 

27.7 

23.2 

20.8 

2O.6 

22.9 

27.7 

35.2 

45.5 

34 

35 

54.3 

44-1 

35.8 

29.3 

24.8 

22-4 

22.3 

24-6 

29.4 

37.0 

47-4 

35 

36 

55.8 

45.7 

37.3 

3o.8 

26.4 

24.I 

24.O 

26.3 

3l.2 

38.8 

49-3 

36 

3? 

57.3 

47.2 

38.8 

32.4 

28.0 

25.7 

25.  7 

28.1 

33.o 

4o.6 

5i.i 

37 

38 

58.8 

48.7 

4o.4 

34.o 

29.6 

274 

27.4 

29.8 

34-8 

42.4 

53.o 

38 

39 

6o.3 

5o.2 

4i.9 

35.6 

31.3 

29.0 

29.1 

3i.5 

36.5 

44.2 

54.9 

39 

4o 

3261.7 

3351.7 

3443.5 

3537.i 

3632.8 

373o.7 

383o.8 

3933.2 

4o38.3 

4i46.i 

4256.7 

4o 

4i 

63.2 

53.2 

45.o 

38.7 

34.4 

32.3 

32.4 

35.o 

4o.i 

47-9 

58.6 

4i 

42 

64-7 

54-7 

46.6 

4o.3 

36.i 

34.o 

34.i 

36.  7 

4i.8 

49-7 

6o.5 

42 

43 

66.2 

56.2 

48.i 

4i.9 

37.7 

35.6 

35.8 

38.4 

43.6 

5i.5 

62.3 

43 

44 

67.7 

57.8 

49-7 

43.5 

39.3 

37.3 

37.5 

40.2 

45.4 

53.4 

64.2 

44 

45 

69.2 

59.3 

5l.2 

45.o 

4o-9 

38.9 

39.2 

41.9 

47.2 

55.2 

66.1 

45 

46 

70.7 

60.8 

52.8 

46.6 

42.5 

4o.6 

40.9 

43.6 

4g.o 

57.0 

68.0 

46 

4? 

72.1 

62.3 

54.3 

48.2 

44-1 

42.2 

42.6 

45.4 

5o.7 

58.8 

69.8 

47 

48 

73.6 

63.8 

55.8 

49.8 

•  45.8 

43-9 

44-3 

47-i 

52.5 

60.7 

71.7 

48 

49 

•75.i 

65.4 

57.4 

5i.4 

474 

45.5 

46.o 

48.8 

54.3 

62.5 

73.6 

49 

5o 

3276.6 

3366.9 

3458.9 

3553.0 

3649-0 

3747.2 

3847-7 

SgSo.e 

4o56.i 

4i64.3 

4275.5 

5o 

5i 

78.1 

68.4 

6o.5 

54.6 

5o.6 

48.8 

494 

52.3 

57.8 

66.1 

77-4 

5i 

5a 

79.6 

69.9 

62.0 

56.i 

52.2 

5o.5 

Si.i 

54-0 

59.6 

68.0 

79-2 

52 

53 

81.1 

71.2 

63.6 

57.7 

53.8 

52.1 

52.8 

55.8 

61.4 

69.8 

81.1 

53 

54 

82.6 

73.o 

65.2 

59.3 

55.5 

53.8 

54-4 

57.5 

63.2 

71.6 

83.o 

54 

55 

84-i 

74-5 

66.7 

60.9 

57.i 

55.5 

56.i 

59.3 

65.o 

73.5 

84-9 

55 

56 

85.6 

76.0 

68.3 

62.5 

58.7 

57.i 

57.8 

61.0 

66.8 

75.3 

86.8 

56 

57 

87.1 

77.5 

69.8 

64.0 

6o.3 

58.8 

59.5 

62.7 

68.5 

77.i 

88.6 

57 

58 

88.5 

79.0 

71.4 

65.6 

61-9 

60.4 

61.2 

64.5 

7o.3 

70.  c 

90.5 

58 

59 

90.0 

80.6 

73.o 

67.2 

63.6 

62.1 

62.9 

66.2 

72.1 

80.8 

92.  z 

59 

MERILIONAL    TARTS. 


147 


LATITUDE. 

Min. 

58° 

59° 

60° 

61° 

62° 

63° 

64° 

65° 

66° 

67° 

68° 

Min. 

o 

4294.3 

4409.1 

4527.4 

4649-2 

4775.0 

4904.9 

5o39.4 

5r7S.8 

5323.5 

5474.0 

563o.8 

0 

i 

06.2 

n.  i 

29.4 

5i.3 

77.1 

07.1 

4i.7 

81.2 

26.0 

76.6 

33.5 

I 

2 

98.! 

i3.o 

3i.4 

53.4 

79.3 

09.4 

44.o 

83.5 

28.4 

79.1 

36.2 

2 

3 

43oo.o 

i5.o 

33.4 

55.4 

81.4 

n.6 

46.3 

85.9 

30.9 

81.7 

38.8 

3 

4 

OI.O 

16.9 

35.4 

57.5 

83.5 

i3.8 

48.6 

88.3 

33.4 

84.3 

4i.5 

4 

5 

o3.? 

18.9 

37-4 

59.6 

85.6 

16.0 

5o.9 

90.7 

35.8 

86.9 

44-2 

5 

6 

o5.6 

20.8 

394 

61.6 

87.8 

18.2 

53.i 

93.1 

38.3 

89.4 

46.9 

e 

7 

oy.S 

22.8 

4i-4 

63.7 

89.9 

20.4 

55.4 

95.4 

4o.8 

92.0 

49.6 

7 

8 

09.4 

24.7 

43.4 

65.8 

92.1 

22.6 

57.7 

97.8 

43.2 

94.5 

52.3 

8 

9 

ii.3 

26.7 

45.4 

67.8 

94.2 

24.8 

60.0 

52OO.2 

45-7 

97.1 

54-9 

9 

10 

43i3.2 

4428.6 

4547-4 

4669.9 

4796.3 

4927.0 

5o62.3 

52O2.6 

5348.2 

5499.7 

5657.6 

10 

ii 

i5.i 

3o.6 

49.4 

72.0 

98.5 

29.2 

64-6 

04.9 

5o.7 

55o2.3 

6o.3 

ii 

12 

17.0 

32.5 

5i.4 

74.1 

48oo.6 

3i.5 

66.9 

07.3 

53.i 

04.9 

63.o 

12 

i3 

18.9 

34-5 

53.5 

76.1 

02.8 

33.7 

69.2 

09.7 

55.6 

07.4 

65.7 

i3 

i4 

20.  8 

36.4 

55.5 

78.2 

o4-9 

35.9 

7i.5 

12.  1 

58.i 

IO.O 

68.4 

i4 

i5 

22.7 

38.4 

57.5 

8o.3 

07.1 

38.i 

73.8 

14-5 

60.6 

12.6 

71.1 

i5 

16 

24.6 

4o.3 

59.5 

82.4 

09.2 

4o.3 

76.1 

16.9 

63.i 

15.2 

73.8 

16 

i? 

26.5 

42.3 

6i.5 

84-5 

n.4 

42.6 

78.4 

19.3 

65.6 

17.8 

76.5 

17 

18 

28.4 

44.2 

63.5 

86.5 

i3.5 

44-8 

80.7 

21.6 

68.0 

20.4 

79.2 

18 

J9 

3o.3 

46.2 

65.6 

88.6 

i5.7 

47.0 

83.o 

24.0 

7o.5 

23.0 

81.9 

J9 

20 

4332.2 

4448.2 

4567.6 

4690.7 

4817.8 

4949.2 

5o85.3 

5226.4 

5373.0 

5525.6 

5684-6 

20 

21 

34.1 

5o.i 

69.6 

92.8 

2O.O 

5i.5 

87.6 

28.8 

75.5 

28.1 

87.3 

21 

22 

36.o 

52.1 

71.6 

94.9 

22.1 

53.7 

90.0 

3l.2 

78.0 

30.7 

90.0 

22 

23 

37.9 

54-1 

73.6 

97.0 

24-3 

55.9 

92.3 

33.6 

8o.5 

33.3 

92.7 

23 

24 

39.8 

56.o 

75.7 

99.1 

26.4 

58.2 

94.6 

36.o 

83.o 

35.9 

95.5 

24 

25 

4i.8 

58.o 

77-7 

4701.1 

28.6 

60.4 

96.9 

38.4 

85.5 

38.6 

98.2 

25 

26 

43-7 

6o.<> 

79-7 

03.2 

3o.8 

62.6 

99.2 

4o.8 

88.0 

41.2 

5700.9 

26 

27 

45.6 

61.9 

81.7 

o5.3 

32.9 

64-9 

5ioi.5 

43.2 

90.5 

43.8 

o3.6 

27 

28 

47-5 

63.9 

83.8 

07.4 

35.i 

67.1 

o3.8 

45.7 

93.0 

46.4 

o6.3 

28 

29 

49.4 

65.9 

85.8 

09.5 

37.3 

69.4 

06.2 

48.x 

95.5 

49.0 

09.1 

29 

3o 

435i.3 

4467.8 

4587.8 

4711.6 

4839.4 

4971.6 

5io8.5 

525o.5 

5398.o 

555i.6 

5711.8 

3o 

3i 

53.2 

69.8 

89.9 

i3.7 

4i.6 

73.8 

10.8 

52.9 

54oo.5 

54-2 

i4.5 

3i 

32 

55.i 

71.8 

91.9 

i5.8 

43.8 

76.1 

i3.i 

55.3 

o3.o 

56.8 

17.3 

32 

33 

57.i 

73.7 

93.9 

17.9 

45.9 

78.3 

i5.5 

57.7 

o5.5 

59.4 

20.0 

33 

34 

59.0 

75.7 

96.0 

2O.O 

48.i 

80.6 

17.8 

60.  i 

08.1 

62.1 

22-7 

34 

35 

60.9 

77-7 

98.0 

22.1 

5o.3 

82.8 

20.1 

62.6 

10.6 

64-7 

25.5 

35 

36 

62.8 

79-7 

46oo.o 

24-2 

52.4 

85.i 

22.4 

65.o 

i3.i 

67.3 

28.2 

36 

37 

64.7 

81.6 

02.1 

26.3 

54-6 

87.3 

24-8 

67.4 

i5.6 

69.9 

30.9 

37 

38 

66.7 

83.6 

o4-i 

28,4 

56.8 

89.6 

27.I 

69.8 

18.1 

72.6 

33.7 

38 

39 

68.6 

85.6 

06.1 

3o.5 

59.0 

91.8 

29.4 

72.2 

20.6 

75.2 

36.4 

39 

4o 

4370.5 

4487.6 

4608.2 

4732.6 

486i.i 

4994.1 

5i3i.8 

5274.7 

5423.2 

5577.8 

5739.2 

4o 

4i 

72.4 

89.6 

IO.2 

34.7 

63.3 

96.3 

34.i 

77.1 

25.7 

80.4 

41.9 

4i 

42 

74-3 

9i.5 

12.3 

36.8 

65.5 

98.6" 

36.5 

79.5 

28.2 

83.i 

44-7 

42 

43 

76.3 

93.5 

i4.3 

39.0 

67.7 

5ooo.8 

38.8 

82.0 

3o.8 

85.7 

47.5 

43 

44 

78.2 

95.5 

16.4 

4i.i 

69.9 

o3.i 

4t.i 

84-4 

33.3 

88.4 

50.2 

44 

45 

80.1 

97.5 

18.4 

43.2 

72.0 

o5.4 

43.5 

86.8 

35.8 

91.6 

52.9 

45 

46 

82.1 

99.5 

2O.5 

45.3 

74-  a 

07.6 

45.8 

89.3 

38.4 

93.6 

55.7 

46 

47 

84.o 

45oi.5 

22.5 

47-4 

76.4 

09.9 

48.2 

91.7 

40.9 

96.3 

58.5 

47 

48 

85.9 

o3.4 

24.6 

49-5 

78.6 

12.2 

5o.5 

94.1 

43.4 

98.9 

61.2 

48 

49 

87.8 

o5.4 

26.6 

5i.6 

80.8 

144 

52.9 

96.6 

46.o 

56oi.6 

64.o 

49 

5o 

4389.8 

45o7.4 

4628.7 

4753.7 

4883.o'5oi6.7 

5:55.2 

5299.0 

5448.5 

56o4.2 

5766.8 

5o 

5i 

91.7 

09.4 

30.7 

55.9 

85.2 

18.9 

57.6 

53oi.5 

5i.o 

06.9 

69.5 

5i 

62 

93.6 

11.4 

32.8 

58.o 

87.4 

21.2 

59.9 

o3.g 

53.6 

09.5 

72.3 

52 

53 

95.6 

1  3.4 

34.8 

60.1 

89.5 

23.5 

62.3 

o6.3 

56.i 

12.2 

75.i 

53 

54 

97.5 

i5.4 

36.9 

62.2 

91.7 

2-5.8 

64.6 

08.8 

58.7 

i4.8 

77-9 

54 

55 

99.4 

17-4 

Sg.o 

64-4 

93.9 

28.0 

67.0 

II.  2 

61.2 

i7.5 

80.6 

55 

56 

44oi.4 

19.4 

4i.o 

66.5 

96.1 

3o.3 

69.4 

13.7 

63.8 

20.2 

83.4 

56 

57 

o3.3 

21.4 

43.o 

68.6 

98.3 

32.6 

71.7 

16.1 

66.3 

22.9 

86.2 

57 

58 

o5.3 

23.4 

45.i 

70.7 

4900.5 

34-9 

74.i 

18.6 

68.9 

25.5 

89.0 

58 

59 

07.2 

25.4 

47.2 

72.9 

02.7 

37.i 

76.4 

21.  1 

7i.4 

28.2 

91.8 

59 

148 


MERIDIONAL    PARTS. 


LATITUDE. 

Min. 

69° 

70° 

71° 

72° 

73° 

74° 

75° 

76° 

77° 

78° 

79° 

Mia. 

o 

5794.6 

5965.9 

6145.7 

6334.8 

65344 

6745.7 

6970.3 

7210.1 

7467.2!7744.6 

8045.7 

O 

I 

97-4 

G8.S 

4«.8 

38.i 

37.8 

49.4 

74.2 

14.2 

7.1.7 

49.4 

5i.o 

I 

2 

58oo.i 

71.8 

5i.9 

41.3 

4i.3 

53.o 

78.1 

i8.3 

76.1 

54.2 

56.2 

2 

3 

02.9 

74-7 

54-9 

44-6 

44-7 

56.6 

81.9 

22.5 

80.6 

59.0 

6i.5 

3 

4 

o5.7 

77.6 

58.o 

47-8 

48.i 

6o.3 

85.8 

26.6 

85.o 

63.9 

66.7 

4 

5 

o8.5 

80.6 

61.1 

5i.i 

5i.6 

63.9 

89.7 

3o.8 

89.5 

68.7 

72.0 

5 

6 

iz.3 

83.5 

64.2 

54-3 

55.o 

67.6 

93.6 

35.o 

94.0 

73.5 

'    77-3 

6  ! 

7 

l4-2 

86.4 

67.3 

57.6 

58.5 

71.2 

97.5 

39.i 

98.5 

78.4 

82.6 

j 

8 

17.0 

89.4 

70.4 

60.8 

61.9 

74-9 

7001.4 

43.3 

75o3.o 

83.3 

87.9 

8 

9 

19.8 

92.3 

73.5 

64.1 

65.3 

78.5 

o5.3 

47-5 

07.4 

88.1 

93.2 

9 

10 

5822.6 

SggS.S 

6176.6 

6367.4 

6568.8 

6782.2 

7009.2 

725i.7 

75n.9 

7793.o 

8o98f5 

10 

ii 

25.4 

98.2 

79-7 

70.6 

72.3 

85.9 

i3.i 

55.8 

16.5 

97-9 

8io3.8 

ii 

12 

28.2 

6001.2 

82.8 

73.9 

75.7 

89.6 

17.0 

60.0 

21.  O 

7802.8 

00.2 

12 

i3 

3i.o 

04.1 

85.9 

77.2 

79-2 

93.2 

20.9 

64.2 

25.5 

07.7 

i4.5 

i3 

i4 

33.8 

07.1 

89.0 

80.4 

82.6 

96.9 

24.8 

68.4 

3o.o 

12.6 

19.9 

*4 

i5 

36.7 

IO.O 

92.1 

83.7 

86.1 

6800.6 

28.8 

72.6 

34.5 

i7.5 

25.2 

i5 

16 

Sg.S 

i3.o 

95.2 

87.0 

89.6 

o4.3 

32.7 

76.8 

39.i 

22.4 

3o.6 

16 

17 

42.3 

16.0 

98.3 

90.3 

93.0 

08.0 

36.6 

81.1 

43.6 

27.3 

36.o 

>7 

18 

45.i 

18.9 

6201.4 

93.6 

96.5 

11.6 

4o.6 

85.3 

48.i 

32.2 

4i-3 

18 

19 

48.o 

21.9 

o4.5 

96.9 

6600.0 

1  5.4 

44-5 

89.5 

52.7 

3?.2 

46.7 

*9 

20 

585o.8 

6024.9 

6207.7 

64oo.2 

66o3.5 

6819.1 

7048.5 

7293-7 

7557.3 

7842.1 

8162.1 

20 

21 

53.6 

27.8 

10.8 

o3.4 

o7.o 

22.8 

52.4 

98.o 

61.8 

47.i 

57.5 

21 

22 

56.5 

3o.8 

18.9 

06.7 

io.5 

26.5 

56.4 

73O2.  2 

66.4 

52.  0 

62.9 

22 

23 

59.3 

33.8 

17.0 

IO.I 

14.0 

30.2 

6o.3 

06.4 

71.0 

57.0 

68.4 

23 

24 

62.2 

36.8 

2O.2 

i3.4 

17.6 

33.9 

64.3 

10.7 

75.5 

6i.9 

73.8 

24 

25 

65.o 

39.8 

23.3 

16.7 

21.  0 

37.6 

68.3 

i5.o 

80.1 

66.9 

79.2 

25 

26 

67.8 

42.7 

26.5 

20.  o 

24.5 

4i.3 

72.2 

I9.2 

84-7 

7i.9 

84-7 

26 

27 

70.7 

45.7 

29.6 

23.3 

28.0 

45.i 

76.2 

23.5 

89.3 

76-9 

90.1 

27 

28 

73.5 

48.7 

32.7 

26.6 

3i.5 

48.8 

80.2 

27.7 

93.9 

8i.9 

95.6 

28 

29 

76.4 

5i.7 

35.9 

29.9 

35.o 

52.5 

84.2 

32.0 

98.! 

86.9 

8201.1 

29 

3o 

5879.2 

6064.7 

6239.0 

6433.3 

6638.5 

6856.3 

7088,2 

7336.3 

7603.2 

789i.9 

8206.6 

3d 

3i 

82.1 

57.7 

42.2 

36.6 

42.1 

60.0 

92.2 

4o.6 

07.8 

96.9 

12.  1 

3i 

32 

85.o 

60.7 

45.3 

39.9 

45.6 

63.8 

96.2 

44.9 

12.4 

7902.0 

I7.6 

32 

33 

87.8 

63.7 

48.5 

43.2 

49.1 

67.5 

7100.2 

49.2 

17.0 

07.0 

23.1 

33 

34 

90.7 

66.7 

5i.7 

46.6 

52.6 

71.3 

O4«2 

53.5 

21.7 

12.  0 

28.6 

34 

35 

93.6 

69.7 

54.8 

49.9 

56.2 

75.o 

08.2 

57.8 

26.3 

I7.I 

34.1 

35 

36 

96.4 

72.7 

58.o 

53.3 

59.7 

78.8 

12.2 

62.1 

3i.o 

22.1 

39.7 

36 

37 

99.3 

75.7 

61.2 

56.6 

63.3 

82.6 

i6.3 

66.4 

35.6 

27.2 

45.2 

37 

38 

5902.2 

78.8 

64-3 

60.0 

66.8 

86.3 

20.3 

70.7 

4o.3 

32.3 

5o.8 

38 

39 

o5.o 

81.8 

67.5 

63.3 

70.4 

90.1 

24.3 

75.i 

45.o 

37.3 

56.3 

39 

4o 

5907.9 

6084.8 

6270.7 

6466.7 

6673.9 

6893.9 

7128.4 

73794 

7649.7 

7942.4 

8261.9 

4o 

4i 

10.8 

87.8 

73.9 

70.0 

77-4 

97-7 

32.4 

83.7 

54.3 

47.5 

67.5 

4i 

42 

i3.7 

90.8 

77.1 

73.4 

81.0 

6901.5 

36.4 

88.1 

59.0 

52.6 

73.i 

42 

43 

16.6 

93.9 

80.2 

76.7 

84.6 

o5.3 

4o.5 

92.4 

63.7 

57.7 

78.6 

43 

44 

19.4 

96.9 

83.4 

80.1 

88.2 

09.1 

44.5 

96.8 

68.4 

62.8 

84-3 

44 

45 

22.3 

99.9 

86.6 

83.5 

91.7 

12.9 

48.6 

74oi.i 

73.2 

68.0 

89.9 

45 

46 

25.2 

6io3.o 

89.8 

86.9 

95.3 

16.7 

52.7 

o5.5 

77-9 

73.1 

95.5 

46 

47 

28.1 

06.0 

93.o 

90.2 

98.9 

20.  5 

56.7 

09.9 

82.6 

78.2 

83oi.i 

47 

48 

3i.o 

09.0 

96.2 

93.6 

6702.5 

24.3 

60.8 

i4.3 

87.3 

83.4 

06.7 

48 

49 

33.9 

12.  1 

99.4 

97.0 

06.  i 

28.! 

64-9 

18.6 

92.1 

88.5 

12.4 

49 

5o 

5936.8 

6n5.i 

6302.6 

65oo.4 

6709.7 

693i.9 

7169.0 

7423.0 

7696.8 

7993.7 

83i8.i 

5o 

5i 

39.7 

18.2 

o5.8 

o3.8 

l3.2 

35.7 

73.i 

27.4 

7701.5 

98.9 

23.8 

5i 

52 

42.6 

21.2 

09.1 

07.2 

16.8 

93.6 

77.2 

3i.8 

06.3 

8oo4.o 

29.4 

62 

53 

45.5 

24.3 

!2.3 

10.6 

20.^! 

43.4 

81.2 

36.2 

Il.l 

09.2 

35.i 

53 

54 

48.4 

27.3 

i5.5 

i4-o 

24.0 

47.2 

85.3 

4o.6 

i5.8 

1  44 

4o.8 

54 

55 

5i.3 

3o.4 

18.7 

17-4 

27.7 

5i.i 

89.5 

45.i 

2o.6 

19.6 

46.5 

55 

56 

54.2 

33.4 

21.9 

20.8 

3i.3 

54.9 

93.6 

49.5 

25.4 

24.8 

62.2 

56 

57 

57.2 

36.5 

25.1 

24.2 

34.9 

58.8 

97-7 

53.9 

30.2 

3o.o 

58.o 

57 

58 

60.1 

39.6 

28.4 

27.6 

38.5 

62.6 

7201.8 

58.3 

35.o 

35.2 

63.7 

58 

59 

63.o 

42.6 

3i.6 

3i.o 

42.1 

66.5 

o5.6 

62.8 

39.8 

4o.5 

69.4 

59 

CORRECTIONS  TO  MIDDLE  LATITUDE. 


143 


Mid. 
Lat. 

1° 

DIFFERENCE  OF  LATITUDE. 

Mid. 
Lat. 

2D 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

0 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

I 

1 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

0 

i5 

o 

2 

3 

5 

7 

9 

12 

i5 

18 

22 

26 

3! 

36 

4i 

47 

52 

59 

65 

72 

i5 

16 

o 

2 

3 

4 

6 

9 

II 

i4 

18 

21 

25 

3o 

34 

39 

44 

5o 

56 

62 

69 

ib 

17 

o 

2 

3 

4 

6 

8 

II 

i4 

J7 

2O 

24 

28 

33 

38 

43 

48 

54 

60 

66 

I7 

18 

0 

I 

3 

4 

6 

8 

10 

i3 

16 

2O 

23 

27 

32 

36 

4i 

46 

52 

58 

64 

18 

J9 

o 

I 

3 

4 

6 

8 

IO 

i3 

16 

19 

22 

26 

3o 

35 

4o 

45 

5o 

56 

61 

J9 

20 

o 

I 

2 

4 

5 

7 

IO 

12 

i5 

18 

22 

25 

29 

34 

38 

43 

48 

54 

60 

20 

21 

o 

I 

2 

4 

5 

7 

9 

12 

i5 

18 

21 

25 

29 

33 

37 

42 

47 

52 

58 

21 

22 

0 

I 

2 

4 

5 

7 

9 

12 

i4 

J7 

21 

24 

28 

32 

36 

4i 

46 

5i 

56 

22 

23 

o 

I 

2 

3 

5 

7 

9 

II 

i4 

I7 

2O 

23 

27 

3i 

35 

4o 

45 

5o 

55 

23 

24 

o 

I 

2 

3 

5 

7 

9 

II 

i4 

16 

20 

23 

27 

3i 

35 

39 

44 

49 

54 

24 

25 

o 

I 

2 

3 

5 

7 

9 

II 

i3 

16 

J9 

23 

20 

3o 

34 

39 

43 

48 

53 

25 

26 

0 

1 

2 

3 

5 

6 

8 

II 

i3 

16 

19 

22 

26 

3o 

34 

38 

42 

47 

52 

26 

27 

0 

I 

2 

3 

5 

6 

8 

II 

i3 

16 

J9 

22 

25 

29 

33 

37 

42 

4? 

52 

27 

28 

o 

I 

2 

3 

5 

6 

8 

IO 

i3 

16 

18 

22 

25 

29 

33 

37 

4i 

46 

5i 

28 

29 

o 

I 

2 

3 

5 

6 

8 

IO 

i3 

i5 

18 

21 

25 

28 

32 

37 

4i 

46 

5i 

29 

3o 

o 

I 

2 

3 

5 

6 

8 

IO 

i3 

i5 

18 

21 

25 

28 

32 

36 

4i 

45 

5o 

3o 

3i 

0 

I 

2 

3 

5 

6 

8 

10 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

5o 

3i 

32 

0 

o 

I 

2 

3 

4 

6 

8 

10 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

5o 

32 

33 

o 

o 

I 

2 

3 

4 

6 

8 

10 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

33 

34 

o 

0 

I 

2 

3 

4 

6 

8 

IO 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

34 

35 

o 

o 

I 

2 

3 

4 

6 

8 

IO 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

35 

36 

0 

I 

2 

3 

4 

6 

8 

10 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

36 

37 

o 

I 

2 

3 

4 

6 

8 

10 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

37 

38 

o 

I 

2 

3 

4 

6 

8 

10 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

5o 

38 

39 

o 

I 

2 

3 

4 

6 

8 

IO 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

5o 

39 

4o 

o 

I 

2 

3 

5 

6 

8 

IO 

13 

i5 

18 

21 

25 

28 

32 

36 

4i 

45 

5o 

4o 

4i  '  o 

I 

2 

3 

5 

6 

8 

10 

l3 

i5 

18 

21 

25 

28 

32 

37 

4i 

46 

5ij  4i 

42 

0 

I 

2 

3 

5 

6 

8 

10 

13 

i5 

18 

22 

25 

29 

33 

37 

4i 

46 

5i  42 

43 

o 

I 

2 

3 

5 

6 

8 

IO 

13 

16 

18 

22 

25 

29 

33 

37 

42 

46 

52  i  43 

44 

o 

I 

2 

3 

5 

6 

8 

IO 

13 

16 

:9 

22 

25 

29 

33 

38 

42 

47 

52 

44 

45 

0 

I 

2 

3 

5 

6 

8 

II 

i3 

16 

J9 

22 

26 

3o 

34 

38 

43 

48 

53 

45 

46 

0 

I 

2 

3 

5 

6 

8 

II 

i3 

16 

J9 

22 

26 

3o 

34 

38 

43 

48 

53 

46 

4? 

o 

I 

2 

3 

5 

7 

9 

II 

i3 

16 

J9 

23 

26 

3o 

35 

39 

44 

49 

54 

47 

48 

o 

I 

2 

3 

5 

7 

9 

II 

i4 

17 

20 

23 

27 

3i 

35 

4o 

44 

5o 

55 

48 

49  ' 

o 

I 

2^ 

3 

5 

7 

9 

II 

i4 

17 

20 

23 

27 

3i 

36 

4o 

45 

5o 

56 

49 

5o 

0 

I 

2 

4 

5 

7 

9 

II 

i4 

17 

20 

24 

28 

32 

36 

4i 

46 

5i 

57 

5o 

5i 

0 

I 

2 

4 

5 

7 

9 

12 

i4 

I7 

21 

24 

28 

32 

37 

42 

47 

52 

58 

5i 

52 

o 

I 

2 

4 

5 

7 

9 

12 

i5 

18 

21 

25 

29 

33 

38 

43 

48 

53 

59 

52 

53 

o 

I 

2 

4 

5 

7 

IO 

12 

i5 

18 

21 

25 

29 

34 

38 

43 

49 

54 

*? 

53 

54 

o 

I 

2 

4 

5 

7 

IO 

12 

i5 

18 

22 

26 

3o 

34 

39 

44 

5o 

56 

6k 

54 

55   o 

I 

2 

4 

6 

8 

10 

i3 

16 

J9 

22 

26 

3i 

35 

4o 

45 

5i 

57 

63 

55 

56 

0 

I 

3 

4 

6 

8 

10 

i3 

16 

J9 

23 

27 

3i 

36 

4i 

46 

52 

58 

65 

53 

57 

o 

I 

3 

4 

6 

8 

IO 

i3 

16 

20 

24 

28 

32 

37 

42 

48 

54 

60 

66 

57 

58 

o 

2 

3 

4 

6 

8 

II 

i4 

17 

20 

24 

28 

33 

38 

43 

49 

55 

61 

68 

58 

59 

o 

2 

3 

4 

6 

8 

II 

i4 

17 

21 

25 

29 

34 

39 

45 

5o 

57 

63 

70 

59 

60 

0 

2 

3 

4 

6 

9 

II 

i4 

18 

22 

26 

3o 

35 

4o 

46 

52 

58 

65 

72 

60 

61 

o 

2 

3 

5 

7 

9 

12 

i5 

18 

22 

26 

3i 

36 

42 

47 

53 

60 

67 

75 

61 

62 

o 

2 

3 

5 

7 

9 

12 

i$ 

J9 

23 

27 

32 

37 

43 

49 

55 

62 

7° 

77 

62 

63 

0 

2 

3 

5 

7 

10 

12 

16 

20 

24 

28 

33 

39 

44 

5i 

57 

64 

72 

80 

63 

64 

o 

2 

3 

5 

7 

10 

13 

16 

20 

24 

29 

34 

4o 

46 

52 

59 

67 

75 

83 

64 

65 

o 

2 

3 

5 

7 

IO 

i3 

I7 

21 

25 

3o 

36 

4i 

48 

54 

62 

69 

78 

86 

65 

66 

0 

2 

3 

5 

8 

II 

i4 

18 

22 

26 

32 

37 

43 

5o 

57 

64 

72 

81 

9° 

66 

67 

o 

2 

4 

6 

8 

II 

i4 

18 

23 

28 

33 

39 

45 

52 

59 

67 

76 

85 

94 

67 

68 

o 

2 

4 

6 

8 

12 

i5 

'9 

24 

29 

34 

4o 

47 

54 

62 

70 

79 

89 

99 

68 

69 

o 

2 

4 

6 

9 

12 

16 

20 

25 

3o 

36 

42 

49 

57 

65 

74 

83 

93 

io4 

69 

70 

o 

2 

4 

6 

9 

13 

16 

21 

26 

32 

38 

44 

52 

60 

68 

78 

88 

98 

no 

70 

7i 

o 

2 

4 

7 

10 

i3 

17 

22 

27 

33 

4o 

47 

55 

63 

72 

82 

93 

io4 

116 

71 

72 

o 

3 

5 

7 

10 

i4 

18 

23 

29 

35 

42 

49 

58 

67 

76 

87 

98 

in 

124 

73 

150       LOGARITHMS   FOR   COMPUTING   COMPOUND   INIEREST. 


In  computing  compound  inter*  st  for  long  periods  of  time, 
the  following  logarithms  to  more  than  six  places. 

it  is  necessary  to  have 

Number. 

Logarithm. 

Number. 

Logarithm. 

.0026 
.oo5o 

.0075 
.OIOO 
.0125 

.0160 
.0176 
.0200 

.0225 
.O25o 

.0275 
.o3oo 
,o325 
.o35o 
.0875 
.o4oo 

.  00108  438i3 
.OO2l6  60618 

.oo324  5o548 

.00432   13738 

.ooSSg  5o3i9 
.00646  60422 
.00753  44i79 
.00860  01718 
.00966  33167 
.01072  38654 
.01178  i83o5 
.01283  72*47 
.01389  oo6o3 
.0149403498 
.01598  8io54 
.01703  33393 

I.o425 
I  .o45o 
.o475 
.o5oo 
.o525 
.o55o 
.o575 
.0600 
.0625 
,o65o 
.o675 
.0700 
.0725 
.0750 
.o775 
.0800 

,0180760636 
.01911  62904 
.02oi5  4o3i6 
.02118  92991 
.02222  2io45 
.02325  24596 
.02428  03760 
.0253o  58653 
.02632  89387 
.02734  96078 
.02836  78837 
.02938  37777 
.oSoSg  73009 
.o3i4o  84643 
.o324i  72788 
.03342  37555 

NUMBERS  OFTEN  USED  IN  CALCULATIONS. 

Circumference  c 
Surface  of  a  spb 
Area  of  a  circle 
Area  of  a  circle 
Capacity  of  a  sp 
Capacity  of  a  sp 
1  —  3.1415926  . 

f  a  circle  to  diameter  i  } 
ere  to  diameter  i  .  .  .  .  > 

3.1415926 

.7853982 
.5235988 
4.1887902 
o.3i83o99 
°.  2957795 
206264".  8 
.oi74533 
.0002909 
.ooooo485 
.00000970 
.ooooi454 
.00001939 
.00002424 
.00002909 
.00003394 
.00003879 
.oooo4363 
2.7182818 
.4342945 
1296000 
864oo 
528o 

Logarithms. 

o.49?i5o 

9.895090 
9.718999 
0.622089 

9.5o285o 
i.758i23 
5.3i44a5 
8.241877 
6.463726 
4-685575 
4.986605 
5.  162696 
5.287635 
5.384545 
5.463726 
5.53o673 
5.588665. 
5.639817 
0.434294 
9.637784 
6.ii26o5 
4-9365i4 
3.722634 

to  radius  i                       ) 

to  diameter  i 

here  to  diameter  i  

, 

here  to  radius  i 



Arc  equal  to  rad 
Arc  equal  to  rad 
Length  of  i  deg 
Length  of  i  min 
Sine  of  i  seconc 
Sine  of  2  second 
Sine  of  3  seconc 
Sine  of  4  seconc 
Sine  of  5  seconc 
Sine  of  6  seconc 
Sine  of  7  seconc 
Sine  of  8  seconc 
Sine  of  9  seconc 
Base  of  Napier' 
Modulus  of  the 
36o  degrees  exf 
24  hours  expres 
Number  of  feet 

ius  expressed  in  degrees 

—5? 

ius  expressed  in  seconds  . 

ree  in  parts  of  radius 



ute  in  parts  of  radius  





•:: 



Is  

, 

Is  



s  



Is 

is".  ..:::::.::: 



Is            .      . 



Is  

._  

s  system  of  logarithms 

;  

common  logarithms  

_ 

>ressed  in  seconds 



sed  in  seconds  

__ 

in  one  mile    .    ... 



THE 

END. 

DR.  LOOMIS'S  MATHEMATICAL  SERIES, 

PUBLISHED  BY 

HARPER  &  BROTHERS,  NEW  YOKK. 


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